Continuity Question

• Mar 29th 2007, 04:11 AM
fifthrapiers
Continuity Question
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.
• Mar 29th 2007, 06:24 AM
ThePerfectHacker
Quote:

Originally Posted by fifthrapiers
Can a function be continuous everywhere but differentiable nowhere?

Yes! (In fact I never go though a week without mentioning it at least once to somebody).

Weierstraß.

Quote:

Also, why is the sum of infinite continuous functions a discontinuous one? Kind of crazy to me.
Why not? We never proven that otherwise.

(Why is a sum of infinite rational numbers rational (further more transendental!!!) same idea).
• Mar 29th 2007, 09:46 AM
AfterShock
Quote:

Originally Posted by fifthrapiers
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.

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.
• Mar 29th 2007, 11:05 AM
ThePerfectHacker
Quote:

Originally Posted by AfterShock
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].