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Integration as Antiderivative

show that eve...

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show that every differential function is continuous

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$Let the statement be f (x) is differentiable at x = c \Rightarrow f (x) is continuous at x = c Proof Let a function f (x) be differentiable at x = c . Then, \lim_{x \to c} \frac{f (x) - f (x)}{x - c} exists finitely . Let \lim_{x \to c} \frac{f (x) - f (x)}{x - c} = f' (c) . . . . (1) In order to prove that f (x) is continuous at x = c, it is sufficient to show that \lim_{x \to c} f (x) = f (c) Now, f (x) = \lim_{x \to c} \{(\frac{f (x) - f (c)}{x - c}) (x - c) + f (c)\} \Rightarrow \lim_{x \to c} f (x) = \lim_{x \to c} \{(\frac{f (x) - f (c)}{x - c}) (x - c)\} + f (c) \Rightarrow \lim_{x \to c} f (x) = \lim_{x \to c} (\frac{f (x) - f (c)}{x - c}) \lim_{x \to c} (x - c) + f (c) = f' (c) \times 0 + f (c) [Using 1] \Rightarrow \lim_{x \to c} f (x) = f (c) Hence, f (x) is continuous at x = c$
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