I have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the concept better. Thanks!
• Pointwise convergent sequences that do / do not preserve continuity,
• Pointwise convergent sequences that do / do not preserve
integrals.
It seems that $f_n(x) = x^n$ is an example that do not preserve continuity?
Best Answer
Here are two useful (non)examples and one somewhat trivial well-behaved example.
Example 1. Take $f_n(x)=\frac{1}{1+x^n}$ for $x\in [0,1]$. By a direct calculation, we see $$ \lim_{n\to\infty} f_n(x)= \begin{cases} 1&x\in [0,1)\\ \frac{1}{2}&x=1. \end{cases}$$ Here, the pointwise limit has a discontinuity. You should try to prove why this is not a uniformly convergent family.
Example 2. Define the characteristic function $$ \chi_{[0,\frac{1}{n}]}(x)= \begin{cases} 1&x\in [0,1/n]\\ 0& x\not\in [0,1/n]. \end{cases}$$ Next. set $f_n(x)=n\chi_{[0,\frac{1}{n}]}$. Then for any $n$, we have $$ \int_0^1 f_n(x)dx=1.$$ However, $\lim_{n\to\infty} f_n(x)=f(x)\equiv 0.$ In particular, we do not have uniform convergence. As a consequence, we can see that $$ \int_0^1\lim_{n\to\infty} f_n(x)dx=\int_0^10\cdot dx=0.$$ So, the limit does not preserve the integral in this case.
Example 3. An easy example of uniform convergence is something like $f_n(x)=\frac{n}{n+1} x$ on $[0,1]$ which is continuous. Clearly, $\lim_{n\to\infty} f_n(x)= f(x)=x$ on $[0,1]$. Furthermore, $$ \lim_{n\to\infty}\int_0^1f_n(x)dx=\lim_{n\to\infty}\int_0^1\frac{n}{n+1}x dx=\lim_{n\to\infty} \frac{n}{n+1}\int_0^1 xdx=\int_0^1f(x)dx.$$