Calculus I · Limits and Continuity · lesson

Continuity on Intervals and at Endpoints

Concept

Learning objectives

Determine intervals of continuity; use one-sided limits at endpoints; state the natural domains on which familiar functions are continuous.

Continuity on Intervals and at Endpoints

A function is continuous on an open interval if it is continuous at every point in the interval.

At the left endpoint $a$ of a closed interval $[a,b]$ , continuity requires

\lim_{x\to a^+}f(x)=f(a).

At the right endpoint $b$ , continuity requires

\lim_{x\to b^-}f(x)=f(b).

Concept

At the left edge of a road, there is no road to the left. It would be silly to demand traffic from a direction that does not exist. Endpoint continuity checks only from inside the interval.

Guided walkthrough

Square root at its endpoint

Is $f(x)=\sqrt{x}$ continuous at $x=0$ on its domain $[0,\infty)$ ?

Show worked solution

The function value is

f(0)=0.

Only the right-hand limit is relevant because negative inputs are outside the real domain:

\lim_{x\to0^+}\sqrt{x}=0.

The right-hand limit equals the function value, so $f$ is continuous at the endpoint $0$ .

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.

Source & rights

Original instruction with traceable references.

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