# Give an example of a function that satisfies strict inequality in Fatou's lemma and …

Find a sequence of functions $f_n\ge 0$ that is continuous on the interval $[0,1]$ satisfies $\int f_n dm \leq 1$ and strict inequality occur in Fatou's lemma.

I could find one that satisfies the strict inequality but discontinuous. example characteristics function $f_n(x)=\large{X}_{[n,n+1]}(x)$. but I need one that is continuous as well and satisfies all the above. any assisstance is appreciated.

• Could you take $X_{[n,n+1]}(x)$ and "round out" the edges so that the integral stays less than 1 but it becomes continuous? – j4l3kl24jkl2 Nov 11 '16 at 15:28
• By 'round out' the edges you mean make the interval an open one? $(n,n+1)$ . – J. Kyei Nov 11 '16 at 15:41
• No I more mean connect the discontinuous points at the edges of the interval in a continuous way such that the integral becomes strictly less than 1. I can post an image if you'd like. – j4l3kl24jkl2 Nov 11 '16 at 15:47
• Kindly post the image for me. – J. Kyei Nov 11 '16 at 16:00
• Note that those characteristic functions are not defined on $[0,1].$ – zhw. Nov 11 '16 at 16:52

You should be able to make $X_[n,n+1]$ continuous by shrinking the interval where your function is exactly one, and connecting the points in a small epsilon neighborhood of $n$ and $n+1$, like so In this way, the integral is $<1$, but the function becomes continuous.
Try the sequence $f_n(x) = n^2x^n(1-x).$