## Differentiable function - Wikipedia, the free encyclopedia

--------------------

** Differentiable function **

A differentiable function

In calculus (a branch of mathematics), a *differentiable function* of one
real variable is a function whose derivative exists at each point in its
domain. As a result, the graph of a differentiable function must have a
(non-vertical) tangent line at each point in its domain, be relatively
smooth, and cannot contain any breaks, bends, or cusps.

More generally, if /x/[] is a point in the domain of a function /f/, then
/f/ is said to be *differentiable at /x/[]* if the derivative /f/
â²(/x/[]) exists. This means that the graph of /f/ has a non-vertical
tangent line at the point (/x/[], /f/(/x/[])). The function /f/ may also
be called *locally linear* at /x/[], as it can be well approximated by a
linear function near this point.

*Contents*

· 1 Differentiability and continuity
· 2 Differentiability classes
· 3 Differentiability in higher dimensions
· 4 Differentiability in complex analysis
· 5 Differentiable functions on manifolds
· 7 References

*Differentiability and continuity*

The absolute value function is continuous (i.e. it has no gaps). It is
differentiable everywhere /except/ at the point /x/ = 0, where it makes a
sharp turn as it crosses the /y/-axis.
An ordinary cusp on the cubic curve (semicubical parabola) /x/^3 â
/y/^2 = 0, which is equivalent to the multivalued function /f/(/x/) = Â±
/x/^3/2. This relation is continuous, but is not differentiable at the
cusp.

If /f/ is differentiable at a point /x/[], then /f/ must also be continuous
at /x/[]. In particular, any differentiable function must be continuous at
every point in its domain. /The converse does not hold/: a continuous

Source: en.wikipedia.org/wiki/Differentiable_function