Differentiable
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A real function 
is said to be differentiable at a point 
if is defined in an open neighborhood of and all partial derivatives of exist at . In particular, for a function defined on some subset 
of 
taking values in , is differentiable at 
if and only if contains an open interval containing and the derivative of exists at .
A function 
can fail to be differentiable at the point 
for the following reasons:
- is not defined at , i.e.
doesn't exist.
- is not defined on some set of points that includes members arbitrarily close to .
- The derivative
is not defined at . Note that this requires at the very least that 
, i.e. any function differentiable at a point must also be continuous at that point.
