Continuity on an interval and of piecewise functions

What you will be able to do

• decide whether a function is continuous on an open or a closed interval;
• use the algebra of continuous functions to find where a formula is continuous;
• find the constants that make a piecewise function continuous.

Continuity on an interval

A function is continuous on an open interval $(a, b)$ when it is continuous at every point of $(a, b)$.

For a closed interval $[a, b]$ the two end points have only one side. So $f$ is continuous on $[a, b]$ when

• it is continuous at every point of $(a, b)$,
• $\displaystyle\lim_{x \to a^+} f(x) = f(a)$, and
• $\displaystyle\lim_{x \to b^-} f(x) = f(b)$.

For example, $f(x) = \sqrt{x}$ is continuous on $[0, \infty)$. At $x = 0$ only the right-hand limit makes sense, and it equals $0 = f(0)$.

Building continuous functions from simple ones

The limit laws give a short list of rules. Suppose $f$ and $g$ are continuous at $a$. Then these are continuous at $a$ too:

• the sum $f + g$, the difference $f - g$, and a constant multiple $cf$;
• the product $fg$;
• the quotient $\dfrac{f}{g}$, provided $g(a) \ne 0$;
• the composite $f(g(x))$, provided $f$ is continuous at $g(a)$.

The standard functions are continuous wherever they are defined.

• Polynomials, $|x|$, $\sin x$, $\cos x$ and $e^x$ are continuous for every real $x$.
• A rational function $\dfrac{p(x)}{q(x)}$ is continuous wherever $q(x) \ne 0$.
• $\ln x$ is continuous for $x > 0$, and $\tan x$ is continuous except at $x = \dfrac{\pi}{2} + n\pi$.
• The inverse trigonometric functions are continuous on their domains.

For example, $|x^2 - 4x|\sin x$ is continuous everywhere. It is built from a polynomial, the absolute value and the sine by composition and products.

The greatest integer function $[x]$ is the exception to remember. It is constant between integers, so it is continuous there, and it jumps at every integer.

A note on words

NCERT calls $f(x) = \dfrac{1}{x}$ a continuous function, because it is continuous at every point of its domain, and $0$ is not in the domain. Many A-Level, IB and AP texts say instead that it has an infinite discontinuity at $x = 0$. Both describe the same graph. They differ on whether a point outside the domain counts. Follow the wording of the question.

Piecewise functions

A piecewise function has a different formula on different intervals. Each piece is a standard function, so it is continuous inside its own interval. Only the joining points need a check.

1. At each joining point $a$, find the left-hand limit from the piece on the left.
2. Find the right-hand limit from the piece on the right.
3. Read off $f(a)$.
4. Set all three equal. Each joining point gives one equation, so two unknown constants need two joining points.

A worked example

A function is defined by $f(x) = 2x + p$ for $x < 2$, $f(2) = 7$, and $f(x) = x^2 - q$ for $x > 2$. Find $p$ and $q$ so that $f$ is continuous at $x = 2$.

1. Left-hand limit. As $x \to 2^-$, $f(x) \to 2(2) + p = 4 + p$. Set $4 + p = 7$. Then $p = 3$.
2. Right-hand limit. As $x \to 2^+$, $f(x) \to 2^2 - q = 4 - q$. Set $4 - q = 7$. Then $q = -3$.
3. Check. The pieces are now $2x + 3$ and $x^2 + 3$. Both give $7$ at $x = 2$.

Practise with new numbers

Try it in Python

The program takes the function of the worked example and tries $p = 0, 1, \dots, 5$. For each $p$ it prints the left-hand limit and the gap between it and the value $7$.

Common mistakes

• Checking every point. Inside a piece the function is already continuous. Check the joining points only.
• Using the wrong piece. The left-hand limit uses the formula for $x$ below $a$. The right-hand limit uses the formula for $x$ above $a$.
• Forgetting the middle value. When $f(a)$ has its own value, as in $f(2) = 7$, all three numbers must agree.
• Ignoring the domain of a quotient. A rational function is discontinuous where its denominator is zero.

Remember these

• Continuous on $[a, b]$ means continuous inside, right-continuous at $a$ and left-continuous at $b$.
• Sums, products, quotients (with a nonzero denominator) and compositions of continuous functions are continuous.
• For a piecewise function, check each joining point: left-hand limit, right-hand limit and value must be equal.
• $[x]$ is continuous between integers and jumps at every integer.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Where is $f(x) = \dfrac{x - 1}{x^2 - 9}$ discontinuous?

2. Let $f(x) = 2x + 1$ for $x \le 2$ and $f(x) = kx$ for $x > 2$. Find $k$ so that $f$ is continuous at $2$.

3. Is $[x]$ continuous on the open interval $(2, 3)$?

4. Define $f(x) = \dfrac{\sin x}{x}$ for $x \ne 0$ and $f(0) = 1$. Is $f$ continuous at $0$?

5. Is $\tan x$ continuous on the closed interval $\left[0, \dfrac{\pi}{2}\right]$?

References

• OpenStax, Calculus Volume 1, Chapter 2, section on continuity.
• NCERT, Mathematics Part I for Class XII, Chapter 5, section on continuity.