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

In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)).

exist and are continuous over the domain of the function

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

In particular, any differentiable function must be continuous at every point in its domain.

The converse does not hold: a continuous function need not be differentiable.

For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meagre set in the space of all continuous functions.

Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.

Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function).

Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem.

Similarly to how continuous functions are said to be of class

continuously differentiable functions are sometimes said to be of class

if the first and second derivative of the function both exist and are continuous.

the function is smooth or equivalently, of class

A function of several real variables f: Rm → Rn is said to be differentiable at a point x0 if there exists a linear map J: Rm → Rn such that If a function is differentiable at x0, then all of the partial derivatives exist at x0, and the linear map J is given by the Jacobian matrix, an n × m matrix in this case.

A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus.

However, the existence of the partial derivatives (or even of all the directional derivatives) does not guarantee that a function is differentiable at a point.

For example, the function f: R2 → R defined by is not differentiable at (0, 0), but all of the partial derivatives and directional derivatives exist at this point.

In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions.

This is allowed by the possibility of dividing complex numbers.

when Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition.

is automatically differentiable at that point, when viewed as a function

can be differentiable as a multi-variable function, while not being complex-differentiable.

is differentiable at every point, viewed as the 2-variable real function

does not exist (the limit depends on the angle of approach).

Such a function is necessarily infinitely differentiable, and in fact analytic.