Title: Questions saved from context Quiz: Review of continuous functions for semester 1.
Course: HELM Workbooks 2023-2024

These questions (some of which may be hidden) were saved when context Quiz: Review of continuous functions for semester 1 was deleted because they are still used by some quizzes or other activities.

Continuous and discontinuous functions and limits

Look at the graph shown in Figure a. 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 Figure 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]

Figure: (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.

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: \(\qquad f(x)= \left\{ \begin{array}{ll} 3 & x<0\\ x^2 & x\ge 0 \, . \\ \end{array} \right.\)

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]

Figure: 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 \(x=a\).

Q: Exercise

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

\(y=x^2\) 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:

\(y=3x-2\) 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]

Q: Exercise

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

\(f(x)= x^3-1\) 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