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)]


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 .

Formal Limit

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)   .