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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.