# What does differentiable mean for a function?

geometrically, the function $f$ is differentiable at $a$ if it has a non-vertical tangent at the corresponding point on the graph, that is, at $\left(a , f \left(a\right)\right)$. That means that the limit
${\lim}_{x \setminus \to a} \frac{f \left(x\right) - f \left(a\right)}{x - a}$ exists (i.e, is a finite number, which is the slope of this tangent line). When this limit exist, it is called derivative of $f$ at $a$ and denoted $f ' \left(a\right)$ or $\frac{\mathrm{df}}{\mathrm{dx}} \left(a\right)$. So a point where the function is not differentiable is a point where this limit does not exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different one-sided limits (a cusp, like for $f \left(x\right) = | x |$ at 0). See definition of the derivative and derivative as a function.