Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.

If x₀ is an interior point in the domain of a function f, then f is said to be differentiable at x₀ if the derivative ${\displaystyle f'(x_{0})}$ exists. In other words, the graph of f has a non-vertical tangent line at the point (x₀, f(x₀)). f is said to be differentiable on U if it is differentiable at every point of U. f is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function ${\textstyle f}$. Generally speaking, f is said to be of class ${\displaystyle C^{k}}$ if its first ${\displaystyle k}$ derivatives ${\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}$ exist and are continuous over the domain of the function ${\textstyle f}$.

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