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.