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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.
Differentiable programming is making significant strides in various fields beyond its traditional applications. In healthcare and life sciences, for example, it is being used for deep learning in biophysics-based modelling of molecular mechanisms.
However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function . [ 15 ] In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions.
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve .
For example, suppose that : is a differentiable function of variables , …,. The total derivative of f {\displaystyle f} at a {\displaystyle a} may be written in terms of its Jacobian matrix, which in this instance is a row matrix:
The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case.
A bump function is a smooth function with compact support.. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain.
A function differentiable at a point is continuous at that point. Differentiation is a linear operation in the following sense: if and are two maps which are differentiable at , and is a scalar (a real or complex number), then the Fréchet derivative obeys the following properties: () = (+) = + ().