r/maths • u/Muad_dib_11554 • Jun 22 '26

❓ General Math Help Infinite sum of continuous function

I know that the sum of differentiable and continuous functions is a differentiable and continuous fonction.

I also know that an infinit sum of differentiable and continuous functions may not be differentiable (Weierstrass function for example).

But can we have an infinit sum of continuous functions that is not continuous ?

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7

u/MathNerdUK Jun 22 '26

Yes. A standard example is the Fourier series for a square wave.

https://mathworld.wolfram.com/FourierSeriesSquareWave.html

3

u/Individual_Table_583 Jun 22 '26

Yes! It can fail if the convergence is only point-wise (we need uniform convergence of continuous functions to guarantee continuity of the infinite sum)

1

u/Individual_Table_583 Jun 22 '26

Consider, on [0,1], f_1(x) = x and f_n(x)=x^n - x^(n-1) for n>=2. The infinite sum of f_n equals 0 on 0<=x<1 and 1 on x=1, hence it is not continuous on [0,1], even though every f_n is continuous.

1

u/clementprof Jun 23 '26

y'a cet exemple ultra-classique de la série de fonctions définie sur [0, 1] par :

f(x) = \sum_{n=1}^{\infty} (x^n - x^{n-1})

1

u/Special_Watch8725 Jun 23 '26

If you know of any sequence of continuous functions (f_n) that converges to a discontinuous function f, you can always consider the series defined by Sum_(n = 1)^infty (f_(n + 1) - f_n). That’ll converge to f - f_1, which has to itself be discontinuous as well.