Example or Counter example of uniform equicontinuous family

Question

I am studying for my final on differential equations and there are some exam problems on real analysis that I really dont know how I even start with.

Suppose that $(g_{n})$ is a sequence equicontinuous and decreasing of real functions defined in $\mathbb{R}$, converging to a function $g:\mathbb{R}\rightarrow\mathbb{R}$.

1) Give an example where the sequence ($g_{n}$) is uniformly equicontinuous, and another one where is not. Are both cases possible?

2) Give an example where g is continuous and another one where it is not. Are both cases possibles?

I don't know if those are really easy questions but the are NO EXAMPLES whatsoever or exercises of any kind on these concepts which are absolutly new to me -it is Msc in applied math. The only material covered on the text book are some definitions and theorems stated without proof or example.

The definition of point equicontinuity and uniform equicontinuity I am working with are as follow:

*Point equicontinuity: the family of functions {${fn}$} is equicontinuous at $x_{0}\in A$ if for any arbitrary $\epsilon>0$, $\exists\ \delta>0 : \vert f_{n}(x)-f_{n}(x_{0}) \vert<\epsilon,\,\, \forall x\in A:\vert x-x_{0}\vert<\delta\quad and \, \forall n\in\mathbb{N}$

*Uniform Equicontinuity: the family of functions {${fn}$} is equicontinuous over A if for any arbitrary $\epsilon>0$, $\exists\ \delta>0 : \vert f_{n}(x)-f(z) \vert<\epsilon,\,\, \forall x\in A:\vert x-z \vert<\delta\quad and \, \forall n\in\mathbb{N}$

Can anyone please (PLEASE!!) give a hand with those?? Thank you! Javi

Answer 1

1. $f_n(x)=x^2+\frac1n$ is equicontinuous, but not uniformly so. $g_n(x)=x+\frac1n$, on the other hand, is uniformly equicontinuous.

2. In both examples above, they converge to a continuous function. An equicontinuous sequence of continuous functions cannot have a discontinuous limit. This is because if a sequence of functions converges pointwise and is equicontinuous, then it converges uniformly.