Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
2  
If this is a homework problem, please tag it as "homework". –  Ron Gordon Dec 31 '12 at 12:28
2  
What have you tried? Which statements do you think are true, and why? –  Asaf Karagila Dec 31 '12 at 12:49
    
i am not sure how to start.give any clue please –  gomti Dec 31 '12 at 12:53
add comment

2 Answers 2

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

$$ |f_n(x_n)-f(x)|=|(f_n(x_n)-f_n(x))+(f_n(x)-f(x))|$$

$$\leq |f_n(x_n)-f_n(x)|+|f_n(x)-f(x)| < \dots.$$

share|improve this answer
add comment

Hints:

(a) Try proving $ |\overline{\int^b_a} f(x) dx-\underline\int_a^b f(x)\,dx|\le 2\epsilon_n(b-a).$ Where $\epsilon_n=$sup$|f_n(x)-f(x)| , $ keep in mind $\epsilon_n \to0 $ as $n\to \infty$

(b) If $f'_n(x)$ is continuous,and if it converges uniformly use $\displaystyle\lim_{x\to\infty}\int^b_af'(x)dx=\int^b_ag(x)$ and show $g(x)=f'(x)$

(c) is straightforward

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.