When Is a Piecewise Function Differentiable?

For a continuous function , small changes in cause small changes in .

There are several ways that a function can be discontinuous at a point . If either of the one-sided limits does not exist, is not continuous. If the one-sided limits both exist but are unequal, i.e., , then has a jump discontinuity.

For a function to be differentiable at a point , it has to be continuous at but also smooth there: it cannot have a corner or other sudden change of direction at . For example, the absolute value function has a sharp turn at 0.

This Demonstration asks you to find the value of to get rid of a jump discontinuity, of to make a continuous function differentiable, or of and to make a function with a jump discontinuity differentiable.