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Blumberg theorem – even if a real function : is nowhere continuous, there is a dense subset of such that the restriction of to is continuous. Thomae's function (also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
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 Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows: , = (( (!))) for integer j and k. This shows that the Dirichlet function is a Baire class 2 function.
Similarly, every additive function that is not linear (that is, not of the form for some constant ) is a nowhere continuous function whose restriction to is continuous (such functions are the non-trivial solutions to Cauchy's functional equation). This raises the question: can such a dense subset always be found?
A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. [1] The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions.
Continuous function: in which preimages of open sets are open. Nowhere continuous function: is not continuous at any point of its domain; for example, the Dirichlet function. Homeomorphism: is a bijective function that is also continuous, and whose inverse is continuous. Open function: maps open sets to open sets.
An example of a Darboux function that is nowhere continuous is the Conway base 13 function. Darboux functions are a quite general class of functions. It turns out that any real-valued function ƒ on the real line can be written as the sum of two Darboux functions. [5]
A function's being odd or even does not imply differentiability, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous. In the following, properties involving derivatives, Fourier series, Taylor series are considered, and these concepts are thus supposed to be defined for the considered functions.