Can a function be continuous everywhere but differentiable nowhere?
Also, why is the sum of infinite continuous functions a discontinuous one? Kind of crazy to me.
Yes! (In fact I never go though a week without mentioning it at least once to somebody).
Weierstraß.
Why not? We never proven that otherwise.Also, why is the sum of infinite continuous functions a discontinuous one? Kind of crazy to me.
(Why is a sum of infinite rational numbers rational (further more transendental!!!) same idea).
For your first question, indeed! I'll even challenge you more. Did you know that a function can be continuous at a POINT- think about rational numbers? Also, fractals are nicely applied here.
For the second question, there exists a function that is discontinuous when adding an infinite number of continuous functions (this is not true for all functions obviously). There's even a proof in complex analysis that shows an infinite number of continuous functions being added together to create a discontinuous one.
Why be so fancy to use complex analysis?
Define the sequence of functions,
f_n(x) = x^n on [0,1]
We can see that,
lim f_n(x) = f(x) (pointwise) where,
f(x) = 0 if x in [0,1) and 1 if x =1.
Note the sequence of functions {f_n(x)} are continous on [0,1].
But their pointwise limit f(x) is not on [0,1].