Let $f_n$ and $f$ be continuous functions on an interval [$a, b$] and assume that $f_n → f$ uniformly on $[a, b]$. Pick out the true statements:
(a) If $f_n$ are all Riemann integrable, then $f$ is Riemann integrable.
(b) If $f_n$ are all continuously differentiable, then $f$ is continuously differentiable.
(c) If $x_n → x$ in [a, b], then $f_n(x_n) → f(x)$.
how should i able to solve this problem .somebody help me please.
Best Answer
Hints:
(a) A function is Riemann integrable if
$\quad(i)$ it is bounded, and
$\quad(ii)$ has a countable set of discontinuities.
You can see also Lebesgue criteria for Riemann integrability.
(c) you need the fact that if $g$ is a continuous, then
$$ x_n \to x \implies g(x_n)\to g(x)\quad \mathbb{as}\quad n\to \infty, $$
and notice that
$$ |f_n(x_n)-f(x)|=|(f_n(x_n)-f_n(x))+(f_n(x)-f(x))|$$
$$\leq |f_n(x_n)-f_n(x)|+|f_n(x)-f(x)| < \dots.$$