
We see that if a function is differentiable at a point, then it must be continuous at that point.

There are connections between continuity and differentiability.

This theorem is often written as its contrapositive:

If $f(x)$ is not continuous at $x=a$, then $f(x)$ is not differentiable at $x=a$.

Thus from the theorem above, we see that all differentiable functions on $\RR$ are continuous on $\RR$. Nevertheless there are continuous functions on $\RR$ that are not differentiable on $\RR$.

Which of the following functions are continuous but not differentiable on $\RR$?
$x^2$ $\lfloor x \rfloor$ $|x|$ $\frac {\sin (x)}{x}$
Can we tell from its graph whether the function is differentiable or not at a point $a$?
What does the graph of a function $f$ possibly look like when $f$ is not differentiable at $a$?