I have some problem about my homework.
Is the sequence of function $f_n:\mathbb R \to \Bbb R$ defined by
$$f_n(x) = cos(n+x) + log(1+\frac{1}{\sqrt{n+2}}sin^2(n^nx))$$equicountinuous? prove or disprove.
But I have checked the textbook, which says
A sequence of functions ($f_n$) in $C^0$ is equicontinuous if
$$\forall \epsilon >0, \exists\delta>0 $$ such that $$|s-t|<\delta, n\in \Bbb N \implies |f_n(s)-f_n(t)|<\epsilon$$For total clarity, the concept might better be labeled uniform equicontinuity, in contrast to pointwise equicontinuity, which requires
$$\forall\epsilon>0\forall x\in [a,b],\exists\delta>0 $$such that $$|x-t|<\delta,n\in\Bbb N \implies|f_n(x)-f_n(t)|<\epsilon$$
Now my question is, what the difference between pointwise-equicontinuous and uniform-equicontinuous? Can I have some examples? And which does it usually means when only mentioned equicontinuous?
Best Answer
Let $I$ be an index set. Often $I=\mathbb{N}$, but in general $I$ needn't even be countable.
A family of functions $\{ f_\alpha \}_{\alpha \in I}$ are:
continuous if each $\varepsilon$,$x$, and $\alpha$ has an appropriate $\delta$.
uniformly continuous if each $\varepsilon$ and $\alpha$ has an appropriate $\delta$ that doesn't depend on $x$
pointwise equicontinuous if each $\varepsilon$ and $x$ has an appropriate $\delta$ which doesn't depend on $\alpha$
uniformly equicontinuous if each $\varepsilon$ has an appropriate $\delta$ which depends neither on $\alpha$ nor on $x$.
In a lot of applications of the concept of equicontinuity, we are working on a compact metric space. On a compact metric space, essentially the exact same proof that works for the Heine-Cantor theorem lets us see that pointwise equicontinuity and uniform equicontinuity are equivalent. I think this causes some authors to define "equicontinuous" to mean what I am calling "uniformly equicontinuous". Your author seems to be adopting this convention.
In your problem you are not on a compact metric space. So while each function in your family is certainly pointwise continuous, you need to worry both about $n$ being a problem and about $x$ being a problem. The first term and the log aren't too big of a deal. However, you should be worried about $\sin^2(n^n x)$: that $n^n$ is creating a lot of oscillation on short length scales when $n$ is large. You need to see whether the division by $\sqrt{n+2}$ is enough to mitigate that.