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A 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.
For example, if f is a function of two variables x and y, ... Product rule: For two differentiable functions f and g, () = +. An operation d with ...
Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. [14] Under mild conditions (for example, if the function is a monotone or a Lipschitz function), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but ...
Bump functions are examples of functions with this property. To put it differently, the class consists of all continuous functions. The class consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable.
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:
In the examples above, these would be functions, magma homomorphisms, group homomorphisms, ring homomorphisms, continuous functions, linear transformations (or matrices), metric maps, monotonic functions, differentiable functions, and uniformly continuous functions, respectively.
Approximation of the smooth-everywhere, but nowhere-analytic function mentioned here. This partial sum is taken from k = 2 0 to 2 500. A more pathological example is an infinitely differentiable function which is not analytic at any point. It can be constructed by means of a Fourier series as follows. Define for all
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. The Weierstrass function has historically served the role of a pathological function, being the first published ...