steady everywhere but nowhere differentiable. The continuity of this function can be demonstrated by the Weierstrass M test. It is true
such as
according to the formula for the geometric series. Therefore, the series converges uniformly according to the Weierstrass M test. The individual partial sums now form a sequence of continuous functions that converge uniformly to . Thus, as such a limit value is continuous.
literature
Herbert Amann and Joachim Escher, Analysis 1 , Birkhäuser, Basel, 2002. (see Theorem V.1.6)
Individual evidence
↑ H. Heuser: Textbook of Analysis Part 1 . Vieweg + Teubner (2009), sentence 105.3, p. 555.
^ EM Stein, R. Shakarchi: Fourier Analysis. An Introduction. University Press Group Ltd (2003), Theorem 3.1, p. 114.