A differentiable function is any function where the derivative exists at every point in its domain. This means that for every point on the graph of the function there exists a non-vertical tangent line. A differentiable function is also sometimes described as being "smooth" and does not contain any breaks or cusps. For a point to be differentiable it also needs to be continous.

Differentiability can also be described as any point where the following limit (the limit definition of the derivative) exists.

Examples

Differentiable Function

In this case is differentiable at as there exists a tangent line for the graph. Alternatively it can be said that the graph looks "smooth".

Non-Differentiable Functions

Sharp Point

In this case is not differentiable at as there is a sharp point at where no tangent line exists.

Cusp

In this case is not differentiable at as there is a cusp.

Jump

In this case is not differentiable at as it is not continuous at .