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.,
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
to make a function with a jump discontinuity differentiable.