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
Best Answer
$f_n(x)=x^2+\frac1n$ is equicontinuous, but not uniformly so. $g_n(x)=x+\frac1n$, on the other hand, is uniformly equicontinuous.
In both examples above, they converge to a continuous function. An equicontinuous sequence of continuous functions cannot have a discontinuous limit.