# equicontinuous

1. ### Equicontinuous

Prove or disprove F_n(x) = \sin nx is equicontinuous I know the definition of equicontinuous at x_0 it says for all \epsilon >0 there exist \delta>0 such that if d ( f(x_0),f(x) ) < \epsilon then d(x_0 , x) < \delta trying if it is equicontinuous at x_0 = 0 Given \epsilon > 0 |...
2. ### sup of equicontinuous is continuous

Let U={f(x)} be a subset of C_0 (the set of continuous functions f:[a,b]->R) that is equicontinuous and bounded. Prove that sup(f(x): feU) is a continuous function of x. Note: I am not sure if the question means U is itself bounded wrt the sup norm so that sup(|f(x)-g(x)| : xeR) < M for some M...
3. ### if family equicontinuous, prove lipschitz

For a function f:R-->R and a>0 set fa(x) = af(x/a). Prove that if the family (fa) a element in (0,infinity) is equicontinuous then f is lipschitz continuous. Okay so i started the proof with a def of equicontinuity and am trying to move towards lipschitz but having trouble
4. ### Can someone give an example on uniformly bounded sequence but not equicontinuous at 0

Can someone give an example on uniformly bounded sequence of continuous functions on B(0,1) but not equicontinuous at 0? Thanks everyone.
5. ### Compactness

Hello, I need help with the following questions: Let B = {f in C([0,1], R) | f' is continuous on (0,1), f(0)=0, and |f'(x)| <= 1}. Show that cl(B) is compact. I'm stuck on this problem. Appreciate anyone's help on this. Thank you in advance. Thanks :)
6. ### equicontinuous and uniformly equicontinuous

hi,, I would like to give me examples to explain the difference between definition of continuous , equicontinuous and uniformly equicontinuous with respect to sequences ??! I would appreciate any help,,thanks