Question: > 
> Let $f:\mathbb R \to \mathbb R$ be continuous. Then which of the
>  following statements implies that $f(0)=0$?
> 
> 
> (A)$\lim\_{n \to \infty}\int\_{0}^{1}f(x)^n dx=0.$
> 
> 
> (B)$\lim\_{n \to \infty}\int\_{0}^{1}f(x/n) dx=0.$
> 
> 
> (C)$\lim\_{n \to \infty}\int\_{0}^{1}f(nx) dx=0.$
> 
> 
> (D) None of the above.
> 
> 
> 

(A)Suppose $f(x)=1-x, \int\_{0}^{1}(1-x)^n dx=-\frac{1}{n+1},$ which converges to zero. I tried to find the counter example for the (B) and (C). I couldn't find. please help me.

Answer: 
(A) Consider the constant function $f(x) = 1/2$.

(B) $\int\_0^1 f(x/n)\, dx = n \int\_0^{1/n} f(y)\, dy \to f(0)$ (by the mean value theorem), hence condition (B) implies $f(0) = 0$.

(C) Consider $f(x) = \cos(2\pi x)$.
B ) $0=\lim\_{n\to \infty} \int\_0^1f(\frac{x}{n})\, dx\\=\lim\_{n\to \infty} f(\frac{c}{n})(1-0) for \, some \, c\in (0,1)\\= \lim\_{n\to \infty}f(\frac{c}{n})\\= f(0) \, since \, f \, is \, continuous .
$