Differentiable function

In mathematical analysis, a real or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is locally approximable by a linear function at each interior point, and does not contain any break, angle, or cusp.

If ${\displaystyle x_{0}}$ is an interior point in the domain of a real function ${\displaystyle f}$, then ${\displaystyle f}$ is said to be differentiable at ${\displaystyle x_{0}}$ if there exists an ${\displaystyle L\in \mathbb {R} }$ such that for all ${\displaystyle \epsilon >0}$, there exists a ${\displaystyle \delta >0}$ such that for all ${\displaystyle x\in (x_{0}-\delta ,x_{0})\cup (x_{0},x_{0}+\delta )}$, ${\displaystyle {\frac {f(x)-f(x_{0})}{x-x_{0}}}\in (L-\epsilon ,L+\epsilon )}$. In other words, the graph of ${\displaystyle f}$ has a non-vertical tangent line at the point ${\displaystyle (x_{0},f(x_{0}))}$. ${\displaystyle f}$ is said to be differentiable on a subset ${\displaystyle U\subseteq \mathbb {R} }$ if it is differentiable at every point in ${\displaystyle U}$. ${\displaystyle f}$ is said to be continuously differentiable if its derivative is also a continuous function over the domain of ${\textstyle f}$.

Continuous functions may be nowhere differentiable in their domain, such as the Weierstrass function. Taking successive antiderivatives of such a function allows one to obtain a function that is differentiable only a finite number of times, the finite number being any positive integer. Given a positive integer ${\displaystyle k}$, ${\displaystyle f}$ is said to be of class ${\displaystyle C^{k}}$ if its first ${\displaystyle k}$ derivatives ${\textstyle f^{\prime },f^{\prime \prime },\ldots ,f^{(k)}}$ exist and are continuous over the domain of ${\displaystyle f}$.

For a multivariable function, as shown here, the differentiability of it is something more complex than the existence of the partial derivatives of it.