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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.
Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example: A rational function is a quotient of two polynomial functions, and is not defined at the zeros of the denominator.
The characteristic function of the rational numbers is nowhere differentiable yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero, ∫ 1 Q ( t ) φ ( t ) d t = 0. {\displaystyle \int 1_{\mathbb {Q} }(t)\varphi (t)\,dt=0.}
Thomae's function (also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers. Weierstrass function – a function continuous everywhere (inside its domain) and differentiable nowhere.
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.
A natural example of a differential field is the field of rational functions in one variable over the complex numbers, (), where the derivation is differentiation with respect to . More generally, every differential equation may be viewed as an element of a differential algebra over the differential field generated by the (known) functions ...
A complex rational function with degree one is a Möbius transformation. Rational functions are representative examples of meromorphic functions. [3] Iteration of rational functions on the Riemann sphere (i.e. a rational mapping) creates discrete dynamical systems. [4] Julia sets for rational maps
In mathematics, the Dirichlet function [1] [2] is the indicator function of the set of rational numbers, i.e. () = if x is a rational number and () = if x is not a rational number (i.e. is an irrational number).