Differentiability Implies Continuity

If a function has a derivative at a specific point, then it must also be continuous at that point. Likewise, this can be extended to all points by saying that if a function is differentiable at all values in its domain, then it is also continuous for all values in its domain.

Now let's prove this rigorously. Suppose that a function $f(x)$ is differentiable at $x = a$. To show that it is continuous, we need to show that $$\lim_{x \to a} f(x) = f(a)$$ or equivalently, $$\lim_{x \to a}(f(x) - f(a)) = 0.$$

This comes directly from the definition of continuity. Now, because a limit only considers values close to the approaching value but not at that value, $x-a \ne 0$, so we can multiply and divide by $x-a$ without having to worry about illegally dividing by $0$.

We have $$\lim_{x \to a}(f(x) - f(a)) = \lim_{x \to a}\frac{f(x)-f(a)}{x-a} \cdot (x-a)$$.

We can evaluate this limit by considering the factors separately. The first factor is the derivative of $f(x)$ at $x = a$, which we are given to be defined. To see why this is equivalent to the derivative, let $x = a+h$. Then the limit is as $a+h$ approaches $a$, which is equivalent to $h$ approaching $0$, and the expression becomes $\frac{f(a+h)-f(a)}{h}$, which by definition is the derivative upon taking the limit. The second factor approaches $0$ as $x$ approaches $a$. Hence our limit evaluates to $f'(a) \cdot 0 = 0$, so $$\lim_{x \to a}(f(x)-f(a)) = 0$$ or $$\lim_{x \to a}f(x) = f(a)$$ proving continuity, as desired.

Great! See here for a more in depth explanation of why the converse is not necessarily true.

Jeffrey Yu gives a nice proof that differentiability implies continuity. Here's some more intuition:

Ok! Here's some intuition for why the first statement is true:

Let's take a function $f(x)$ suppose that we know $f(x)$ has a derivative of $1$ at $x=0.$ We want to explain why $f(x)$ is continuous at $x=0.$

One example is, $e^x$.

But a straight line is continuous! So...intuitively if $f(x)$ looks like a straight line at $x=0$ (i.e. differentiable!) then it should also be continuous!

(Video) Proof that Differentiability Implies Continuity

by MathemAddicts

Here's a video by MathemAddicts explaining the proof of differentiability implies continuity.