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Question
Is every differentiable function continuous?
Solution
Yes, if a function is differentiable at a point then it is necessary continuous at that point.
\[\text{Proof : Let a function } f(x) \text { be differentiable at x = c . Then}, \]
\[ \lim_{x \to c} \frac{f(x) - f(c)}{x - c}\text { exists finitely } . \]
\[\text { Let } \lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c)\]
\[\text{In order to prove that f(x) is continous at x = c , it is sufficient to show that} \lim_{x \to c} f(x) = f(c)\]
\[ \lim_{x \to c} f(x) = \lim_{x \to c} \left\{ \left( \frac{f(x) - f(c)}{x - c} \right)\left( x - c \right) + f(c) \right\}\]
\[ \Rightarrow \lim_{x \to c} f(x) = \lim_{x \to c} \left[ \left\{ \frac{f(x) - f(c)}{x - c} \right\}\left( x - c \right) \right] + f(c)\]
\[ \Rightarrow \lim_{x \to c} f(x) = \lim_{x \to c} \left\{ \frac{f(x) - f(c)}{x - c} \right\} . \lim_{x \to c} \left( x - c \right) + f(c)\]
\[ \Rightarrow \lim_{x \to c} f(x) = f'(c) \times 0 + f(c)\]
\[ \Rightarrow \lim_{x \to c} f(x) = f(c)\]
\[\text{Hence, f(x) is continuous at x} = c .\]
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