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A real function $f$ is said to be differentiable at a point $P$ if $f$ is defined in an open neighborhood of $P$ and all partial derivatives of $f$ exist at $P$. In particular, for a function $f$ defined on some subset $D$ of $\displaystyle \mathbb{R}$ taking values in $\mathbb{R}$, $f$ is differentiable at $P \in D$ if and only if $D$ contains an open interval containing $P$ and the derivative of $f$ exists at $P$.

A function $f: \mathbb R \to \mathbb R$ can fail to be differentiable at the point $\displaystyle x_0$ for the following reasons:

  • $f$ is not defined at $\displaystyle x_0$, i.e. $\displaystyle f(x_0)$ doesn't exist.
  • $f$ is not defined on some set of points that includes members arbitrarily close to $\displaystyle x_0$.
  • The derivative $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ is not defined at $\displaystyle x_0$. Note that this requires at the very least that $\lim_{h \to 0}f(x_0 + h) = f(x_0)$, i.e. any function differentiable at a point $\displaystyle x_0$ must also be continuous at that point.

See also