Differentiable functions are continuous

Builds on Differentiable function Continuity at a point

Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ , and let $a \in I$ . If $f$ is differentiable at $a$ , then $f$ is continuous at $a$ .

Remarks

For $x \ne a$ , write $f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)$ . The first factor tends to $f'(a)$ and the second to $0$ . The converse fails. The absolute value is continuous but not differentiable at $0$ . The Weierstrass function is continuous and nowhere differentiable. Mathlib states it as HasDerivAt.continuousAt.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Derivative definition
Differentiable function definition
Differentiable functions are continuous statement

Tag	`07ZMX4ANJS`
URI	`m2://07ZMX4ANJS`
Kind	statement, proposition
Writing	W0 stub
MSC	26A24
Status	This copy of the site has no build data. Run `m2 status 07ZMX4ANJS`.
Uses	Differentiable function, Continuity at a point

Differentiable functions are continuous

Statement

Remarks

Reading path

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Further reading

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