Differentiability Implies Continuity

A theorem in Calculus states that if f is differentiable at x = c , then f is continuous at x = c . With this theorem in mind it can be proven that f is continuous at x = c by showing that f(x) approaches f(c) as x \rightarrow c . To do this, we can use the differentiability of f at x = c and consider the following limit.

\displaystyle\lim_{x\to\ c} [f(x) - f(c)] = \displaystyle\lim_{x\to\ c} \bigg[(x - c)\bigg( \frac{f(x) - f(c)}{x - c} \bigg) \bigg]

= \bigg[\displaystyle\lim_{x\to\ c} (x - c) \bigg] \bigg[\displaystyle\lim_{x\to\ c} \frac{f(x) - f(c)}{x - c}\bigg]

= (0)[f\prime(c)]

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