# 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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In this post I am going to introduce the formal definition of a limit. I am still trying to figure out the symbol formatting so bear with me. If $f(x)$ becomes arbitrarily close to a single number $L$ as $x$ approaches $c$ from either side, then the limit of $f(x)$ as $x$ approaches $c$ is $L$, which can be written as $\displaystyle\lim_{x\to\ c} f(x) = L$. This definition may seem formal but isn’t because exact meanings have not yet been given to the two phrases  “ $f(x)$ becomes arbitrarily close to $L$” and “ $x$ approaches $c$“. In the figure below we can let $\epsilon$ represent a (small) positive number. Then the phrase $"f(x)$ becomes arbitrarily close to $L"$ means that $f(x)$ lies in the interval $(L - \epsilon, L + \epsilon)$. Using absolute value, we can write this as $|f(x) - L| < \epsilon$. Similarly, the phrase “ $x$ approaches $c$” means that there exists a positive number $\delta$ such that $x$ lies in either the interval $(c - \delta, c)$ or the interval $(c, c + \delta)$. This fact can be concisely expressed by the double inequality $0 < |x -c| < \delta$. The first inequality $0 < |x -c|$ states that the distance between $x$ and $c$ is more than 0, and expresses the fact that $x \neq c$. The second inequality $|x - c| < \delta$ says that $x$ is within $\delta$ units of $c$. This brings us to a formal definition which can be stated as following: Let $f$ be a function defined on an open interval containing $c$ (except possibly at $c$) and let $L$ be a real number. The statement $\displaystyle\lim_{x\to\ c} f(x) = L$ means that for each $\epsilon > 0$ there exists a $\delta > 0$ such that if $0 < |x-c| < \delta$, then $|f(x) - L| < \epsilon$. This can be expressed symbolically as $\forall \epsilon > 0 \exists \delta > 0 : \forall x (0 < |x - c| < \delta \Rightarrow |f(x) - L| < \epsilon)$  .