Is $(f_n)$ pointwise convergent

real-analysissequence-of-functionsequences-and-series

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. $f_nā†’0$ 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 $2/n ā†’0$ . 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.

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