Let $f_n(x)$, for all n>=1, be a sequence of non-negative continuous functions on [0,1] such that
$$\lim_{nā\infty}\int^1_0 f_n (x)dx=0$$
Which of the following is always correct ?
A. $f_nā0$ uniformly on $[0,1]$
B. $f_n$ may not converge uniformly but converges to $0$ pointwise
C. $f_n$ will converge point-wise and limit may be non zero
D. $f_n$ is not guaranteed to have pointwise limit.
I can find example where all four statements are true but can't find any counterexample.
Option A is certainly false. Consider a function (geometrically) a isosceles triangle whose one vertex is on $0$, other on $2/n$. And height $1$ unit. Its area is $2/n ā0$ . But $f$ is not uniformly continuous
For others, i am completely stuck. I need ideas to solve this question rather than solution.please help!.
Best Answer
A. You could also consider $f_n(x) = x^n.$
B. Let $f_n$ be the line-segment graph connecting the points $(0,1), (1/n,0), (1,0).$ This sequence does not converge to $0$ at $x=0.$
C. Let $f_n$ be as in B. Then consider the sequence $f_1,0,f_2,0,f_3,0,\dots.$ At $x=0,$ this sequence is $1,0,1,0,\dots.$
D. See C.