In a proof of uniform convergence, how to find $\epsilon$ that works for all interval $[a,b]$?

Question

I'm not really getting uniform convergence in terms of actual applications, so I would appreciate some help.

The way my teacher taught uniform convergence, the only example he used was to show a function converges to another function and minimized the Remainder to do so. If the resulting remainder (set less than epsilon) was rearranged to $n$ and still depended on $x$, then he would maximize the function with respect to $x$ and use that value to get an $N$($\epsilon$).

I'm not really how to show it over an interval $[a,b]$.

I've seen problems like this, but I guess my problem is I don't understand where the epsilon comes from. It seems even more arbitrary here than it did when I was doing limits.

Can someone illustrate how to apply this concept to something like $\sin(x)$ over the interval $[-\pi, \pi]$?

(Note: I found a similar Stack Exchange answer, but our teacher doesn't "like" Stirling's Formula, so I'm looking for some additional help)

Answer 1

I don't think you begin to learn about uniform convergence from many concrete examples where you need to find an $N(\epsilon)$ that works for all $x$. That would, in most cases, just be a test of your algebra skills, not a contribution to an understanding of the concept.

Instead focus on problems that illustrate why the concept is important and how it differs from pointwise convergence in significant ways.

Try these problems.

1. Find a sequence of continuous functions $f_n:[0,1]\to\mathbb{R}$ that converges pointwise but not uniformly to zero.

2. Find a sequence of continuous functions $f_n:[0,1]\to\mathbb{R}$ that converges pointwise to a discontinuous function.

3. Find a sequence of continuous functions $f_n:[0,1]\to\mathbb{R}$ that converges pointwise to an unbounded function.

4. Find a sequence of continuous functions $f_n:[0,1]\to\mathbb{R}$ that converges pointwise to a continuous function $f$ but $\int_0^1f_n(x)\,dx\not\to \int_0^1f(x)\,dx$.

5. Find a sequence of continuously differentiable functions $f_n:[0,1]\to\mathbb{R}$ that converges uniformly to a function that is not everywhere differentiable.

6. Find a sequence of continuously differentiable functions $f_n:[0,1]\to\mathbb{R}$ that converges uniformly to a differentiable function but $f_n'\not\to f'$.

7. Find a sequence of step functions $f_n:[0,1]\to\mathbb{R}$ that converges to $f(x)=x$. Is the convergence uniform for your example?

8. Find a sequence of continuous functions $f_n:[0,1]\to\mathbb{R}$ that converges to the step function $f(x)=1$ for $0\leq x\leq \frac12$ and $f(x)=2$ for $\frac12<x\leq 1$. Is the convergence uniform for your example?

Etc.