PlanetMath: differentiable functions are continuous

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differentiable functions are continuous

(Theorem)

Proposition 1 Suppose is an open interval on $\mathbb{R}$ , and $f\colon I\to \mathbb{C}$ is differentiable at $x\in I$ . Then is continuous at . Further, if is differentiable on , then is continuous on .

Proof. Suppose $x\in I$ . Let us show that $f(y)\to f(x)$ , when $y\to x$ . First, if $y\in I$ is distinct to

, then

$\displaystyle f(x)-f(y) = \frac{f(x)-f(y)}{x-y} (x-y).$

Thus, if

is the derivative of

, we have

$\displaystyle \lim_{y\to x} f(x)-f(y)$	$\displaystyle =$	$\displaystyle \lim_{y\to x} \frac{f(x)-f(y)}{x-y} (x-y)$
	$\displaystyle =$	$\displaystyle \lim_{y\to x} \frac{f(x)-f(y)}{x-y}\ \lim_{y\to x} (x-y)$
	$\displaystyle =$	$\displaystyle f'(x)\ 0$
	$\displaystyle =$	$\displaystyle 0,$

where the second equality is justified since both limits on the second line exist. The second claim follows since

is continuous on

if and only if

is continuous at

for all $x\in I$ . $\qedsymbol$

"differentiable functions are continuous" is owned by matte.

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Attachments:: function differentiable at only one point (Example) by matte

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Cross-references: line, limits, equality, derivative, continuous, continuous at, differentiable, open interval
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This is version 5 of differentiable functions are continuous, born on 2004-09-09, modified 2004-10-21.
Object id is 6154, canonical name is DifferentiableFunctionsAreContinuous.
Accessed 2539 times total.

Classification:

AMS MSC:	57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings)
	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)

Pending Errata and Addenda

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