Title: Default for 2.4 Characterising Functions
Course: HELM Workbooks 2023-2024

Continuous and discontinuous functions and limits

Look at the graph shown in part (a) of the figure below. The curve can be traced out from left to right without moving the pen from the paper. The function represented by this curve is said to be continuous at every point. If we try to trace out the curve in part (b), the presence of a jump in the graph (at \(x=x_1\)) means that the pen must be lifted from the paper and moved in order to trace the graph. Such a function is said to be discontinuous at the point where the jump occurs. The jumps are known as discontinuities.

[Image: a) A continuous function; b) A discontinuous function]
(a) A continuous function   (b) A discontinuous function

Try for yourself: Explain the distinction between a continuous and a discontinuous function. Draw a graph showing an example of each type of function.

Try for yourself: Sketch a graph of a function which has two discontinuities.

Sketch a graph of a function which has two discontinuities.

(Get your tutor to check it.)

When defining a discontinuous function algebraically it is often necessary to give different function rules for different values of \(x\). Consider, for example, the function defined as:

\[f(x)= \begin{cases} 3 & x < 0 \\ x^2 & x \geq 0 \end{cases}\]

Notice that there is one rule for when \(x\) is less than 0 and another rule for when \(x\) is greater than or equal to 0.

A graph of this function is shown in the figure below.

[Image: A discontinuous function]
An example of a discontinuous function

Suppose we ask ‘to what value does \(y\) approach as \(x\) approaches 0?’. From the graph we see that as \(x\) gets nearer and nearer to 0, the value of \(y\) gets nearer to 0, if we approach from the right-hand side. We write this formally as

\(\qquad \lim_{x\rightarrow 0^+} f(x) = 0\)

and say ‘the limit of \(f(x)\) as \(x\) tends to 0 from above is 0.’

On the other hand if \(x\) gets closer to zero, from the left-hand side, the value of \(y\) remains at 3. In this case we write

\(\qquad \lim_{x\rightarrow 0^-}f(x) = 3\)

and say ‘the limit of \(f(x)\) as \(x\) tends to 0 from below is 3.’

In this example the right-hand limit and the left-hand limit are not equal, and this is indicative of the fact that the function is discontinuous.

In general a function is continuous at a point \(x=a\) if the left-hand and right-hand limits are the same there and are finite, and if both of these are equal to the value of the function at that point. That is

Key Point

A function \(f(x)\) is continuous at \(x=a\) if and only if: \[\qquad \lim_{x\rightarrow a^+} f(x) = \lim_{x\rightarrow a^-} f(x) = f(a)\]

If the right-hand and left-hand limits are the same, we can simply describe this common limit as \(\lim_{x\rightarrow a} f(x)\). If the limits are not the same we say the limit of the function does not exist at \(x=a\).

Exercise

Explain the distinction between a continuous and a discontinuous function. Draw a graph showing an example of each type of function.

Q: Exercise

Study the graphs of the functions \(y=x^2\) and \(y=-x^2\), \(x \in \mathbb{R}\) and fill in the blanks with the most descriptive valid option:

\(y=x^2\) is [...] and

\(y=-x^2\) is [...].

Note: after checking the answer you will need to drag the items out of the boxes before allowing you to drag back in for the next attempt.
Also offered: discontinuous; neither continuous nor discontinuous; continuous at all but two points
A: continuous; continuous

Q: Exercise

Study the graphs of the following functions defined for all \(x \in \mathbb{R}\) and fill in the blanks with the most descriptive valid option:

\(y=3x-2\) is [...] and

\(y=-7x+1\) is [...].
Also offered: discontinuous; neither continuous nor discontinuous; continuous at all but two points
A: continuous; continuous

Exercise

Draw a graph of the following function for yourself:

\(\qquad f(x)= \left\{ \begin{array}{ll} [value] & x<[value]\\ [value] & x= [value] \\ [value] & x> [value] \end{array} \right.\)

Find the following limits (if a limit does not exist type in none).

1. \(\lim_{x\rightarrow 0^+}f(x) = \)
2. \(\lim_{x\rightarrow 0^-}f(x) = \)
3. \(\lim_{x\rightarrow 0}f(x) = \)
4. \(\lim_{x\rightarrow [value]^+}f(x) = \)
5. \(\lim_{x\rightarrow [value]^-}f(x) = \)
6. \(\lim_{x\rightarrow [value]}f(x) = \)
Feedback: The graph of \[\qquad f(x)= \left\{ \begin{array}{ll} [value] & x<[value]\\ [value] & x= [value] \\ [value] & x> [value] \end{array} \right.\] is shown below [value] We use the graph to find the limits. 1. There is no discontinuity at \(x = 0\), so \(\lim_{x\rightarrow 0^+}f(x) = [value]\). 2. Similarly, \(\lim_{x\rightarrow 0^-}f(x) = [value]\). 3. Similarly, \(\lim_{x\rightarrow 0}f(x) = [value]\). 4. The function is discontinuous at \(x = [value]\). We see from the graph that \(f\) approaches [value] as \(x\) approaches [value] from the right. Therefore, \(\lim_{x\rightarrow [value]^+}f(x) = [value]\). 5. On the other hand, we see that \(f\) approaches [value] as \(x\) approaches [value] from the left. Therefore, \(\lim_{x\rightarrow [value]^-}f(x) = [value]\). 6. Since the one-sided limits of \(f\) at \(x = [value]\) are different, the limit at \(x = [value]\) does not exist.

Periodic functions

Any function that has a definite pattern repeated at regular intervals is said to be periodic. The interval over which the repetition takes place is called the period of the function, and is usually given the symbol \(T\). The period of a periodic function is usually obvious from its graph.

The figure below shows a graph of a periodic function with period \(T=3\). This function has discontinuities at values of \(x\) which are divisible by \(3\).

[Image: A discontinuous periodic function]

The figure below shows a graph of a periodic function with period \(T=6\). This function has no discontinuities.

[Image: A continuous periodic function]

If a function is a periodic function with period \(T\) then, for any value of the independent variable \(x\), the value of \(f(x+T)\) is the same as the value of \(f(x)\).

Key Point

A function \(f(x)\) is periodic if we can find a number \(T\) such that

\(\qquad f(x+T)=f(x)\qquad\text{for all values of}\ x\)

Often a periodic function will be defined by simply specifying the period of the function and by stating the rule for the function within one period. This information alone is sufficient to draw the graph for all values of the independent variable.

The figure below shows a graph of the periodic function defined by

\(\qquad f(x) = x, \quad -\pi<x<\pi, \qquad \hbox{ period } T=2\pi\).

[Image: A periodic function with period 2*pi]

Try for yourself: Sketch a graph of a periodic function which has no discontinuities.

Try for yourself: Sketch a graph of a periodic function which has discontinuities.

Exercise

Explain what is meant by a periodic function.

Exercise

Sketch a graph of a periodic function which has no discontinuities.

Exercise

Sketch a graph of a periodic function which has discontinuities.

Exercise

A periodic function has period \({\displaystyle[value]}\) seconds. How many times will the pattern in the graph repeat over an interval of [value] seconds?

\( N = \)
Feedback: The periodic function has period \({\displaystyle[value]}\) seconds. The frequency (the number of repetitions per time unit) is \( f = \frac{1}{T} = [value]\, \textrm{s}^{-1} = [value]\, \textrm{Hz}.\) So in the time interval of \( [value] \,\textrm{s}\) the pattern repeats \( N = [value]\, \textrm{s}^{-1} \times [value] \, \textrm{s} = [value]\) times.

Odd and even functions

Example

The following figure shows graphs of several functions. They share a common property.

Study the graphs and comment on any symmetry.

[Image: Even functions]

Solution

The graphs are all symmetrical about the \(y\) axis.

Any function which is symmetrical about the \(y\) axis, i.e. where the graph of the right-hand part is the mirror image of that on the left, is said to be an even function. Even functions have the following property:

Key Point

Even Function

An even function is such that \(f(-x)=f(x)\) for all values of \(x\).

This Key Point is saying that the function value at a negative value of \(x\) is the same as the function value at the corresponding positive value of \(x\).

Example

Show algebraically that \(f(x) =x^4+5\) is an even function.

Solution

We must show that \(f(-x)=f(x)\).

\(\qquad f(-x)= (-x)^4 + 5= x^4 + 5\)

Hence \(f(-x)=f(x)\) and so the function is even. Check for yourself that \(f(-3)=f(3)\).

Extend the graph below in order to produce a graph of an even function.

[Image: block6-fig5]
Feedback: [Image: block6-fig5(ans)]

The following diagrams shows  graphs of several functions. They share a common property. Study the graphs and comment on any symmetry.

[Image: block6-fig7]
Feedback: There is rotational symmetry about the origin. That is, each curve, when rotated through \(180^{\circ}\), transforms into itself.

Any function which possesses such symmetry \(-\) that is the graph of the right can be obtained by rotating the curve on the left through \(180^\circ\) about the origin \(-\) is said to be an odd function. Odd functions have the following property:

Key Point

Odd Function

An odd function is such that \(f(-x)=-f(x)\) for all values of \(x\).

This Key Point is saying that the function value at a negative value of \(x\) is minus the function value at the corresponding positive value of \(x\).

Example

Show that the function \(f(x)=x^3+4x\) is odd.

Solution

We must show that \(f(-x)=-f(x)\). \[\begin{aligned} f(-x) &= (-x)^3+4(-x) \\ &= -x^3 - 4x \\ &= - (x^3+4x) \\ &= -f(x)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\end{aligned}\] and so this function is odd. Check for yourself that \(f(-2)=-f(2)\).

Extend the graph below in order to produce a graph of an odd function.

[Image: block6-fig5]
Feedback: [Image: block6-fig5(new)]

Note that most functions are neither odd nor even; for example \(f(x)=x^3+x^2\) is neither even nor odd.

The reader should confirm (with simple examples) that ‘odd’ and ‘even’ functions have the following properties:

\[\begin{aligned} \text{odd}\ +\ \text{odd} &= \text{odd}\qquad \text{even}\ +\ \text{even} = \text{even}\qquad \text{odd}\ +\ \text{even} = \text{neither}\\ \text{odd}\ \times\ \text{odd} &= \text{even}\qquad \text{even}\ \times\ \text{even}\ =\ \text{even}\qquad \text{odd}\ \times\ \text{even}\ =\ \text{odd}\end{aligned}\]

Q: Exercise

Classify the following functions as odd, even or neither. If necessary sketch a graph to help you decide.

1. \(f(x)=6\) is [...]
2. \(f(x)=x^2\) is [...]
3. \(f(x)=2x+1\) is [...]
4. \(f(x)=x\) is [...]
5. \(f(x)=2x\) is [...]
Also offered: even; even; odd; odd; neither odd nor even; neither odd nor even
A: even; even; neither odd nor even; odd; odd

Q: Exercise

The diagram below represents a heavy cable hanging under gravity from two points at the same height. Such a curve (shown as a dashed line), known as a catenary, is described by a mathematical function known as a hyperbolic cosine, \(f(x)=\cosh x\).

[Image: block6-fig6]

Figure: A catenary

1. Comment upon any symmetry: the function is [...].
2. Is this function one-to-one or many-to-one?: [...]
3. Is this a continuous or discontinuous function?: [...]
4. State \(\lim_{x\rightarrow 0} \cosh x\): [...]
Also offered: odd, symmetric about the y-axis; odd, symmetric about the x-axis; odd, rotational symmetry about (0,0); odd, rotational symmetry about (0,1); even, no symmetries; odd, no symmetries; even, symmetric about the x-axis; one-to-one; even, rotational symmetry about (0,0); discontinuous; even, rotational symmetry about (0,1); 0; x; cos(x); -infinity; infinity
A: even, symmetric about the y-axis; many-to-one; continuous; 1

Exercise

Which of the following functions are odd?
Feedback: A function is odd if \(f(-x) = -f(x)\) for all \(x\). This means that the graph of the function for \(x<0\) is obtained by reflecting the graph for \(x>0\) across the \(y\)-axis and then across the \(x\)-axis.

Exercise

Which of the following functions are even?
Feedback: A function is even if \(f(-x) = f(x)\) for all \(x\). This means that the graph of the function for \(x<0\) is obtained just by reflecting the graph for \(x>0\) across the \(y\)-axis. The graph is symmetric across the \(y\)-axis.

Exercise

1. Give an example of an odd function. \(f_1(x)=\) .
2. Give an example of an even function. \(f_2(x)=\) .
3. Give an example of a function which is odd and even. \(f_3(x)=\) .
4. True or false: the answer to 3. is unique.
Feedback: A function \(y=f(x)\) is odd if \(f(-x) = -f(x)\). A function \(y = f(x)\) is even if \(f(-x) = f(x)\). 1. For example, \(f_1(x) = x^3\) is an odd function, since \[f_1(-x) = (-x)^3 = -x^3 = -f_1(x).\] 2. For example, \(f_2(x) = x^4\) is an even function, since \[f_2(-x) = (-x)^4 = x^4 = f_2(x).\] 3. A function that is both even and odd is \(f_3(x) = 0\), since \[f_3(-x) = 0 = f_3(x) = -f_3(x).\] 4. The function \(f_3(x) = 0\) is indeed the unique function that is both even and odd. To see this, let \(f\) be a function that is even and odd and \(x\) be any point on the real line. Then since \(f\) is even we have \(f(-x) = f(x)\), and since \(f\) is odd, \(f(-x) = -f(x)\). Therefore, we must have \(f(x) = -f(x)\), which forces \(f(x) = 0\).

The following diagrams shows graphs of several functions. They share a common property. Study the graphs and comment on any symmetry.

[Image: block6-fig7]

Answer: There is rotational symmetry about the origin. That is, each curve, when rotated through \(180^{\circ}\), transforms into itself.

Q: Exercise

The following diagrams shows  graphs of several functions. They share a common property. Study the graphs and comment on any symmetry.

[Image: block6-fig7]
- There is rotational symmetry about the origin. (Correct. That is, each curve, when rotated through \(180^{\circ}\), transforms into itself.)
- There is reflectional symmetry about the \(x\)-axis. (Incorrect. There is rotational symmetry about the origin. That is, each curve, when rotated through \(180^{\circ}\), transforms into itself.)
- There is reflectional symmetry about the \(y\)-axis. (Incorrect. There is rotational symmetry about the origin. That is, each curve, when rotated through \(180^{\circ}\), transforms into itself.)
- They are identical after scaling centred at the origin. (Incorrect. There is rotational symmetry about the origin. That is, each curve, when rotated through \(180^{\circ}\), transforms into itself.)
A: There is rotational symmetry about the origin.

Q: Exercise

Study the graphs of the following functions for \(x \in \mathbb{R}\) and fill in the blanks with the most descriptive valid option:

\(f(x)= x^3-1\) for \(x<1\) and \(=x^2-1\) for \(x \geq 1\) is [...] and

\(f(x)= x^3-1\) for \(x<-1\) and \(=x^2-1\) for \(x \geq -1\) is [...].
Also offered: neither continuous nor discontinuous; continuous at all but two points
A: continuous; discontinuous