Continuity: Grade 10 Optional Math

A motorbike’s speedometer says 40 km/h. But speed is distance divided by time, and at one instant no time passes and no distance is covered: 0 ÷ 0. So what is the needle showing?

Drop a ball and it falls $s = 5t^2$ meters in t seconds. From t = 2 to a little later, t = 2 + h, its average speed is $\frac{5(2 + h)^2 - 20}{h} = 20 + 5h$ m/s: 20.5 m/s over a tenth of a second, 20.05 m/s over a hundredth, 20.005 m/s over a thousandth. The averages head for 20 m/s, and that is the speed at the instant t = 2, the number the needle shows. Put h = 0 straight into the fraction and you get 0/0, which means nothing; watch where the values head as h shrinks, and you get the answer.

That idea, the value a quantity heads for, is a limit, and this unit’s question is: when does a graph have no break at all, and how can you tell from its formula? A function whose values head for its own value at every point is continuous: its graph can be drawn without lifting the pen.

Close-up of a motorcycle speedometer, its needle pointing to a speed on a dial marked in km/h — **Figure 12.1** A speedometer shows the speed at each instant: what the average speed heads for over shorter and shorter times. Photo: Shixart1985, CC BY 2.0

For the SEE

Limits and continuity carry 7 of the 75 marks: a 1-mark choice, a 4-mark question in group B (a function in pieces tested at one point) and 2 marks inside a group C question.

f(x) is continuous at x = c when all three hold: the limit $\lim_{x \to c} f(x)$ exists (the left-hand and right-hand limits are equal), f(c) is defined, and $\lim_{x \to c} f(x) = f(c)$ .
Left-hand limit (LHL), $\lim_{x \to c^-} f(x)$ : put x = c − 0.1, c − 0.01, c − 0.001 and see what f(x) heads for. Right-hand limit (RHL), $\lim_{x \to c^+} f(x)$ : x = c + 0.1, c + 0.01, c + 0.001.
A function in pieces: the left limit from the piece for x < c, the right limit from the piece for x > c, f(c) from the piece that includes x = c.
$f(c) = \frac{0}{0}$ means f(c) is not defined: discontinuous, even when the limit exists.
To find k for continuity, make the two pieces meet: LHL = RHL = f(c).

How the SEE asks it, with a 4-mark question worked · Exam practice

12.1 Introduction

A curve that has no break anywhere in an interval is a continuous curve on that interval. You can trace it from one end to the other without lifting your pen. A curve with a break, a gap, a jump or a missing point, is discontinuous there.

The idea is old, but it took more than two thousand years to say exactly what it means.

A story from maths

From flowing quantities to three conditions

The Greek philosopher Aristotle (384–322 BCE) discussed the continuous in his Physics: a line or a stretch of time that can be divided again and again without end, never into separate atoms. It was part of his answer to Zeno’s puzzles about motion.

In the 1660s Isaac Newton pictured a quantity as something that flows in time, like the distance covered by a moving body. He called such a quantity a fluent and its rate of flow at an instant its fluxion: the speed the speedometer shows. His calculus is built on quantities that change without jumps.

In the 1700s Leonhard Euler called a function “continuous” when one single formula gave it everywhere, and a function made of pieces “mixed”. That isn’t today’s meaning: x + 2 for x ≤ 2 and 4 − x after it are two formulas but join without a break.

The modern meaning came in the 1800s. Bernard Bolzano in Prague (1817) and Augustin-Louis Cauchy in Paris, in his Cours d’analyse (1821), defined it by the values: a function is continuous when a very small change in x always makes only a very small change in f(x). Karl Weierstrass in Berlin made it exact later in the century, and it comes to the three conditions of this unit: the left and right limits and the value are all defined and all equal.

Continuity is everywhere in science and daily life: the temperature through a day, the height of a growing plant, the water in a filling tank change without jumps. Things counted in whole steps do jump: the number of students in a class, the coins in a pocket, the runs on a cricket scoreboard.

12.2 Continuity and discontinuity

Figure 12.2 What each one touches on the ground, seen from the side: the snake one unbroken stretch, the frog only its landing points.

Figure 12.3 Three figures: a, a circle; b, an X shape drawn as its outline; c, two separate rectangles.

So a and b are continuous figures and c is discontinuous.

A figure that can be drawn from beginning to end without lifting the pen is continuous. If the pen must be lifted somewhere and put down elsewhere, the figure is discontinuous.

Which of these changes continuously?

Pick an answer to see why.

Figure 12.4 Top: the integers are separate dots with gaps between them. Bottom: the real numbers fill the line, with no gap anywhere.

Natural numbers, whole numbers and integers do not form a continuous line; the real numbers do. That is why the graphs in this unit are drawn on the real numbers: “continuous from −5 to 5” means for every real x from −5 to 5, not only the whole numbers.

Worked examples

Exercise 12.1

Question 1

A curve that can be drawn without lifting the pen is called:

Pick an answer to see why.

Which of these may show continuity in daily life?

Pick an answer to see why.

Which points of the number line do the real numbers from −2 to 2 cover?

Pick an answer to see why.

For a function to be continuous at x = a, its graph:

Pick an answer to see why.

Questions 2 to 5

What is meant by a continuous curve and a discontinuous curve? Explain with examples.

Answer

What does it mean to say that a curve is continuous from −5 to +5? Explain with a diagram.

Answer

From the graphs, write the intervals where each graph is continuous and the points where it is discontinuous.

Figure 12.8 Graph a.

Figure 12.9 Graph b.

Figure 12.10 Graph c. The dashed line x = 3 is not part of the graph: the curve climbs beside it without reaching it.

Answer

The wiring of a house is drawn as three graphs.
Wire A is straight and has no break, so current flows through it all the way.
Wire B is bent sharply, at 90°, at a corner of the room (x = 3), but is not broken.
Wire C is broken exactly 4 m along, at x = 4, which stops the current.

Figure 12.11 Wire A.

Figure 12.12 Wire B, with its sharp bend at x = 3.

Figure 12.13 Wire C, broken at x = 4.

a.Is wire A continuous or discontinuous? Give a reason.
b.Is wire B continuous or discontinuous at x = 3? Give a reason.
c.In mathematical language, what is the broken part of wire C at x = 4 called?
d.Compare wire B (a sharp bend) and wire C (a broken wire). In which case is the function (the current’s path) continuous, and in which discontinuous? Why does bending a wire not break its continuity?

Answer

Project

Ages in your neighborhood

Ask the ages of the people living in your neighborhood and fill in the table.

Age group (years)	0–20	20–40	40–60	60–80	above 80
Number of people

From the data, draw the less-than and more-than cumulative frequency curves (ogives). Write a short report on where the curves are continuous or discontinuous, and present it in class.

What to look for

12.3 Continuity and discontinuity of a function

Activity

2: three roads

A road is in three conditions in three places.
Condition A: the road is smooth and unbroken from beginning to end.
Condition B: the road has a sharp V-shaped bend, but is not broken anywhere.
Condition C: an earthquake has broken the road exactly 4 km along (x = 4), leaving a gap.

Figure 12.16 Condition A.

Figure 12.17 Condition B.

Figure 12.18 Condition C: the road y = x − 1, with its gap at x = 4.

a.What kind of graph does condition A make?
b.Compare conditions B and C. Which is a continuous function and which a discontinuous one? Explain.
c.A car is driving along road C. How would you show the break at x = 4 on the graph?
d.If the builders fill the gap at x = 4 level with the road on both sides, explain this with $\lim_{x \to 4} f(x)$ and the value f(4).

What you should see

Limits from the left and from the right

Activity 2 used the main idea of this unit. To see what happens at x = c, you don’t look at c itself first. You look at f(x) for x close to c, and closer and closer, from each side.

The left-hand limit (LHL) is the number f(x) heads for (or approaches) as x comes towards c from below (x = c − 0.1, c − 0.01, c − 0.001, …). It is written $\lim_{x \to c^-} f(x)$ , read “the limit of f(x) as x tends to c from the left”.
The right-hand limit (RHL) is the number f(x) heads for as x comes towards c from above (x = c + 0.1, c + 0.01, …), written $\lim_{x \to c^+} f(x)$ .
When the two are equal, that number is the limit of f(x) at c, written $\lim_{x \to c} f(x)$ . When they differ, the limit does not exist.

The limit says nothing about f(c) itself: x gets as close to c as you like but never equals it. A graph can head for a point that isn’t there.

From these activities: a function f(x) is continuous at x = c when all three conditions hold.

The limit exists: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$ , so $\lim_{x \to c} f(x)$ exists.
The value f(c) is defined.
The limit and the value are equal: $\lim_{x \to c} f(x) = f(c)$ .

If any one of them fails, the function is discontinuous at x = c. In one line: LHL = RHL = f(c).

Bring the points closer to the dashed line: where do their heights head?

f(x) = x + 1, at x = 2

How close: 1 either side of x = 2

From the left: ● f(1) = 2
From the right: ● f(3) = 4

✓ LHL = RHL: the limit exists
✓ f(2) = 3 is defined
✓ the limit equals f(2)

Both sides head for 3, and f(2) = 3. Continuous at x = 2.

Each kind of break fails a different condition. A hole fails condition 2 (the limit exists, the value doesn’t). A moved point fails condition 3 (both exist but differ). A jump fails condition 1 (the two sides don’t agree, so there is no limit).

$\lim_{x \to 5^-} f(x) = 8$ , $\lim_{x \to 5^+} f(x) = 8$ and f(5) = 6. Which is true at x = 5?

Pick an answer to see why.

Worked examples

Joining two pieces: finding k

The SEE often gives a function in two pieces with an unknown k and asks for the k that makes it continuous. Continuity at the joint means the left piece ends exactly where the right piece starts: LHL = RHL = f(c). Each side gives an expression in k; setting them equal gives an equation for k.

Exercise 12.2

Question 1

Which limit of f(x) is $\lim_{x \to a^-} f(x)$ ?

Pick an answer to see why.

Which condition must hold for the limit of f(x) to exist at x = c?

Pick an answer to see why.

What is the value of f(x) = 3x at x = 1?

Pick an answer to see why.

Which condition is needed for f(x) to be continuous at x = a?

Pick an answer to see why.

If f(x) = x + 1, what is $\lim_{x \to 1} \frac{1}{f(x)}$ ?

Pick an answer to see why.

Questions 2 to 10

What do $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$ mean? Explain with examples.

Answer

Write the three conditions for a function f(x) to be continuous at x = a.

Answer

g(x) = x + 3.

a.Find g(x) when x = 2.9, 2.99, 2.999.
b.Find g(x) when x = 3.1, 3.01, 3.001.
c.Find g(3).
d.Find $\lim_{x \to 3^+} g(x)$ and $\lim_{x \to 3^-} g(x)$ .
e.Is g(x) continuous at x = 3? Give a reason.

Answer

h(x) = 2x − 1.

a.Find h(x) when x = 0.9, 0.99, 0.999.
b.Find h(x) when x = 1.1, 1.01, 1.001.
c.Find h(1).
d.Find $\lim_{x \to 1^+} h(x)$ and $\lim_{x \to 1^-} h(x)$ .
e.Is h(x) continuous at x = 1? Give a reason.

Answer

$k(x) = \begin{cases} 3 - x, & 0 \le x \le 1 \\ x + 1, & x > 1 \end{cases}$

a.Find k(x) when x = 0.99.
b.Find k(x) when x = 1.01.
c.Find $\lim_{x \to 1^-} k(x)$ and $\lim_{x \to 1^+} k(x)$ .
d.Is k(x) continuous at x = 1? Verify.

Answer

$p(x) = \begin{cases} 2x + 1, & 1 \le x \le 3 \\ x + 4, & x > 3 \end{cases}$

a.Find p(x) when x = 2.99.
b.Find p(x) when x = 3.01.
c.Find $\lim_{x \to 3^-} p(x)$ and $\lim_{x \to 3^+} p(x)$ .
d.Check whether p(x) is continuous at x = 3.

Answer

Test the continuity of f(x) by putting in values of x:

a.f(x) = 2x + 5 at x = 1
b. $f(x) = \frac{x^2 - 4}{x - 2}$ at x = 2
c. $f(x) = \frac{x^3 - 27}{x - 3}$ at x = 3

Hint

Answer

a. $f(x) = \begin{cases} 7x - 2, & x < 3 \\ 5, & x = 3 \\ 6x + 1, & x > 3 \end{cases}$
Test the continuity of f(x) at x = 3.
b. $g(x) = \begin{cases} x + 2, & 0 \le x \le 1 \\ 3, & x > 1 \end{cases}$
Test the continuity of g(x) at x = 1.

Answer

10.

$f(x) = \begin{cases} 2x - 3, & x < 2 \\ 2k + 3, & x = 2 \\ x + k, & x > 2 \end{cases}$
Find k so that f(x) is continuous at x = 2.

The textbook gives f(2) = 2k − 3. With it the question doesn’t work: LHL = RHL needs 1 = 2 + k, so k = −1, but then f(2) = −5, not 1. With 2k + 3 all three are 1.

Hint

Answer

How the SEE asks it

In the SEE’s Optional Mathematics paper (75 marks, 3 hours), Limits and continuity is a content area of its own. CDC’s specification grid gives it 7 marks, in three kinds of question:

Group	Kind of question	Marks	Levels
A	multiple choice, 1 mark	1	understanding 1
B	within the content area: one function tested at one point, in four 1-mark parts	4	knowledge 1, understanding 1, application 1, higher ability 1
C	across content areas: parts of a longer question joining continuity with another area	2	application 1, higher ability 1

In CDC’s model paper, the 1-mark question gives a function whose left and right limits are equal but whose value at the point is not defined, and asks what follows (it is discontinuous). The group B question is a function in two pieces joined at x = 3, in four 1-mark parts: define continuity, find the limit as x approaches 3, find the value at 3, and say whether it is continuous there, with the reason. In group C, a quadratic function from another area is asked about: is it continuous for every value of its variable, and does a value someone worked out for it at a point fit (check it by working it out yourself).

What each kind asks, and where its marks are:

Knowledge: the three conditions, or the meaning of $\lim_{x \to c^-}$ , $\lim_{x \to c^+}$ , continuous. Write them in symbols: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$ .
Understanding: f(c) from the piece that includes c, or values of f(x) close to c from the piece that applies (2.99 uses the left piece, 3.01 the right).
Application: LHL and RHL, each from its own piece.
Higher ability: the conclusion with its reason, comparing all three numbers; or finding k.
A polynomial (x + 1, $(x + 2)^2 + 5$ , $10k^2 - 76k + 178$ ) is continuous at every point: its LHL, RHL and value are all just the formula’s value. Show it with the values anyway.

Here is a question in the style of group B, written for this book, with how it is marked.

Miscellaneous exercise

Within the content area

g(x) = x + 3.

a.What does it mean for a curve to be continuous or discontinuous at a point? Explain with examples.
b.What is meant by saying that the curve is continuous from −4 to +4?
c.Is g(x) continuous at x = 3? Give a reason.

Answer

f(x) = 4x + 1.

a.Find f(1.999) and f(2.001).
b.Find $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$ .
c.Find f(2).
d.Is f(x) continuous at x = 2? Give a reason.

Answer

$p(x) = \begin{cases} x + 5, & x \le 3 \\ 4x - 4, & x > 3 \end{cases}$

a.When is a function f(x) continuous at x = a? Write it in symbols.
b.Find p(x) when x = 2.9999.
c.Find p(x) when x = 3.0001.
d.Is p(x) continuous at x = 3? Give a reason.

Answer

$h(x) = \begin{cases} kx + 3, & x \le 1 \\ 2x - 1, & x > 1 \end{cases}$

a.What is a continuous function?
b.Find $\lim_{x \to 1^-} h(x)$ and $\lim_{x \to 1^+} h(x)$ .
c.If h(x) is continuous at x = 1, find k.

Answer

Across content areas

These questions join continuity with the other units, as the SEE’s group C questions do.

In the graph, △ABC is the solution region of three inequalities: y ≤ 3, y ≥ x + 1 and y ≥ −2x − 5.

Figure 12.25 The region of y ≤ 3, y ≥ x + 1 and y ≥ −2x − 5: the triangle ABC.

a.Find the maximum and minimum values of 4x − 5y over the points that satisfy all three inequalities.
b.Find the angle between the lines BC and AC.
c.If y = f(x) = x + 1, find f(2.999) and f(3.001).
d.Is f(x) continuous at x = 3? Give a reason.

Answer

$f(x) = \begin{cases} 4x + 3, & x \le 2 \\ 5x + 1, & x > 2 \end{cases}$

a.Does $f^{-1}(11)$ exist? If it does, find it.
b.Test the continuity of f(x) at x = 2.

Answer

In a continuous series, the standard deviation is y, the coefficient of standard deviation is 5x − 1, and the mean is 10.

a.Find y in terms of x.
b.If y = f(x), find $f^{-1}(2)$ .
c.Find the equation of the line through (2, −3) perpendicular to the line y = 5x − 1.

Answer

g(x) = 5x + 2.

a.Find g(2.999) and g(3.001).
b.Is g(x) continuous at x = 3? Give a reason.
c.If y = g(x) is a straight line, find the equation of the line parallel to it through (2, −3).
d.Find $g^{-1}(x)$ .

Answer

The quadratic equation $3x^2 + 5x + 2 = 0$ is given.

a.Solve it graphically.
b.Its factors are 3x + 2 and x + 1; let them be f(x) and g(x). If $g \circ f^{-1}(x) = 2$ , find x.
c.Find the equation of the line through (−3, −2) perpendicular to the line y = 3x + 2.

Answer

AC, AB and BC are three chords of a circle, with A(4, 3) and C(6, −3). The equation of the chord AB is −x + 3y − 5 = 0.

The textbook gives the chord BC as −x + 3y + 5 = 0, but C(6, −3) is not on that line (−6 − 9 + 5 = −10), and with a right angle at C the diameter would be AB, not BC: the book’s numbers don’t work. With the chord AB as here, everything fits.

Figure 12.27 One such circle: with B(10, 5), the circle with diameter BC has its center M(8, 1) and radius

\sqrt{20}

, and passes through A.

a.Find the equation of the chord AC.
b.Find the angle between the chords AC and AB.
c.Prove that BC is a diameter of the circle.
d.Write the equations of AB and AC as y = f(x) and y = g(x) respectively, and find g ∘ f(2).

Answer

The vertices of △ABC are A(−4, 6), B(−6, −10) and C(12, −8). R₁ is the reflection in the line y = 0 and R₂ the reflection in the line x − y = 0.

a.Find the acute angle between the two lines of R₁ and R₂.
b.Find the image of △ABC under the combined transformation R₂ ∘ R₁.
c.Find the scalar product of AB and BC.

Answer

AB is the height of a house. From points C and D on the same level ground, the angles of elevation of the top of the house are 45° and 30°, and CD = 10 m.

Figure 12.28 The house AB and the two points of observation.

a.Define the angle of elevation.
b.Find the height AB.
c.Find the scalar product of DA and DB.
d.What is the geometrical meaning of DA · DB?

Answer

AD is a tower 80 m high. From two points B and C on the same level ground, on opposite sides of the tower, the angles of elevation of its top are both 45°.

Figure 12.29 The tower AD and the two observers at B and C.

a.Define the angle of depression.
b.Find the distance between the two observers.
c.Prove by the vector method that the midpoint of the hypotenuse of a right triangle is equidistant from its three vertices.

Answer

10.

The vertices of △ABC are A(2, 1), B(4, 2) and C(1, 3).

a.Find the matrix T of a rotation of +90° about the origin.
b.Find its transpose $T^{\mathsf{T}}$ .
c.Using $T^{\mathsf{T}}$ , find the image of △ABC.
d.Is the inverse of T defined? Give a reason.

Answer

11.

The vertices of △PQR are P(5, 1), Q(−4, 1) and R(5, 6), with $\angle P = 90°$ ; S is the midpoint of QR.

Figure 12.30 △PQR, right-angled at P, with S the midpoint of QR.

a.If the slopes of PQ and PR are m₁ and m₂, what is the relation between them?
b.Find the scalar product of PQ and PR.
c.Find the equation of the line QR.
d.Prove by the vector method that S is equidistant from the vertices of △PQR.

Answer

12.

The inequalities 3x + 2y + 4 ≥ 0, 3y ≤ 6 and x ≤ 0 are given.

a.Write the equations of the boundary lines.
b.Show the inequalities on a graph and find the solution region.
c.Find any one angle of the polygon formed by the solution region.
d.Using the matrix $T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \end{pmatrix}$ , find the images of the region’s vertices.

Answer

13.

In △ABC, A(1, 3), B(5, 7) and C(7, 3); D is the midpoint of BC.

a.If the position vectors of A and B are a and b, what is the position vector of the midpoint of AB?
b.Find the position vector of D.
c.Using the scalar product, find the angle between AD and BC.
d.Find the equation of AD.

Answer

14.

The position vectors of the vertices A, B and C of a triangle are 2i + 3j, 6i + 7j and 8i + 3j.

a.Find the equation of the median from A to BC.
b.Find the angle between this median and the side BC.

Answer

15.

$f(x) = (x + 2)^2 + 5$ .

a.Sketch its graph.
b.Is the graph continuous or discontinuous? Give a reason.
c.Are $\lim_{x \to 3^-} f(x)$ , $\lim_{x \to 3^+} f(x)$ and f(3) equal? Give a reason.

Answer

Worksheet to print (PDF, 4 pages) Answer key for teachers

Harder worksheet (PDF, 3 pages) Answer key for teachers

Quiz yourself on this unit →

Practice every question in the book, and keep track of the ones you miss →

Where these facts come from

Aristotle’s discussion of the continuous, as what is divisible without end, in his Physics, against Zeno’s paradoxes (12.1): Wikipedia, “Continuous function”, History, and “Zeno’s paradoxes”, read October 2026.
Newton’s fluents and fluxions, from 1665 (12.1): Wikipedia, “Fluxion”, read October 2026.
Euler’s “continuous” functions, given by a single expression, and “mixed” or “discontinuous” ones (12.1): Wikipedia, “History of the function concept”, read October 2026.
Bolzano’s definition (1817), Cauchy’s in the Cours d’analyse (1821) by infinitely small increments, Weierstrass’s epsilon–delta form (12.1): Wikipedia, “Continuous function”, History, and “Cours d’analyse”, read October 2026.
A falling body’s distance $s = \frac{1}{2}gt^2$ (with g taken as 10 m/s², $s = 5t^2$ ), and speed at an instant as the limit of average speeds (the opening): Wikipedia, “Free fall” and “Derivative”, read October 2026.
The photos: their Wikimedia Commons file pages.
CDC’s specification grid and model paper for Optional Mathematics, 2082 (How the SEE asks it): the textbook folder, Specification Grids/grid-Opt. Math-G10.

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