QUESTION: Consider a real valued continuous function $f$ satisfying $f(x+1) = f(x)$ for all $x ∈ \mathbb{R}$. Let
$$g(t)=\int_{0}^tf(x)dx$$ $$h(t)=\lim_{n→\infty}\frac{g(t+n)}n, t∈\mathbb{R}$$ Then--
$(A)$ $h(t)$ is defined only for $t = 0.$
$(B)$ $h(t)$ is defined only when $t$ is an integer.
$(C)$ $h(t)$ is defined for all $t ∈ \mathbb{R}$ and is independent of $t.$
$(D)$ none of the above is true.
Note that this question has been asked once before but I wish to know how to do it using the approach I have done as shown below..moreover I could not complete this from the hint given before..
MY ANSWER: This is what I have done-
Method 1:
Simply using Leibniz integral rule we can say that $g'(t)=f(t)$. Now coming to the limit, using the L'Hôpital's rule we can say that $$h(t)=\lim_{n→\infty}f(t+n)$$ Now, we know $f(x+1)=f(x)$. Therefore $f$ is periodic with period $1$. But, my problem is how do I know the value at infinity?
Since $\infty$ is not a number I don't know the value of $f(t+\infty)$. I guess I am missing on something, or could not utilize the periodic information given correctly..
Method 2:
$$g(t+n)=\int_{0}^{t+n}f(x)dx$$ Now I cannot write it as $(t+n)\int_{0}^1f(x)dx$ since $(t+n)$ is not an integer. So I write $g(t+n)$ as- $$[t+n]\int_{0}^{1}f(x)dx+\int_{0}^{\{t+n\}}f(x)dx$$ where $[t+n]$ denotes the integer part of $(t+n)$ and $\{t+n\}$ denotes the fractional part of $(t+n)$. Now $\int_{0}^1f(x)dx$ is clearly a constant, so no issues in that. But the problem here is again with $n$. As $n→\infty$ how do I solve for $g(t+n)$ ? The fractional part of $\infty$ is not defined, neither (I guess) is the integer part. How do I tackle this problem then?
Any help will be much appreciated. Thank you.
So $$\begin{align}\lim_{n\to\infty}\frac{g(n+t)}n&=\lim_{n\to\infty}\frac{[n+t]\cdot g(1)+g(\{n+t\})}n\\ &=\lim_{n\to\infty}\frac{n\cdot g(1)+[t]\cdot g(1)+g(\{t\})}n\\ &=g(1)+([t]\cdot g(1)+g(\{t\}))\lim_{n\to\infty}\frac{1}n\\ &=g(1)\end{align} $$