Calculus I · Limits and Continuity · reference

Which Functions Are Continuous?

Families of continuous functions

The following functions are continuous everywhere in their natural domains:

• polynomials; • rational functions where the denominator is nonzero; • root functions where the root is real; • trigonometric functions wherever defined; • exponential functions; • logarithmic functions for positive arguments.

Sums, differences, products, constant multiples, and valid compositions of continuous functions are continuous. Quotients are continuous where their denominators are nonzero.

Worked example

Find intervals of continuity

Find the intervals on which

f(x)=\frac{\sqrt{x+2}}{x^2-9}

is continuous.

Show worked solution

The square root requires

x+2\ge0,

x\ge-2.

The denominator must be nonzero:

x^2-9\ne0

x\ne-3,3.

The value $-3$ is already outside the square-root domain. The only excluded domain point within $[-2,\infty)$ is $3$ .

Therefore, the function is continuous on

\boxed{[-2,3)\cup(3,\infty)}.

At $-2$ , continuity is interpreted from the right.

Worked example

Composition

Where is

g(x)=\ln(4-x^2)

continuous?

Show worked solution

The logarithm requires a positive argument:

4-x^2>0.

This means

x^2<4,

-2<x<2.

The polynomial $4-x^2$ is continuous everywhere, and $\ln u$ is continuous for $u>0$ . Therefore the composition is continuous on

\boxed{(-2,2)}.

After the explanation

Use the section idea

Reading lens

Do the limit, the function value, and the surrounding domain fit together at the point or across the interval?

Mental model

Continuity is a three-part agreement: the value exists, the two-sided limit exists, and those two quantities are equal.

Decision

At a point, test the three conditions in order; on an interval, check the domain and endpoints before invoking any continuity theorem.

Common trap

A sign change supports the Intermediate Value Theorem only when continuity holds on the entire closed interval, and it does not prove uniqueness.

Check yourself

You are ready to continue when you can classify a break, decide whether one value can repair it, and state every IVT hypothesis aloud.

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