I have been stuck for several days on this old Analysis problem (I am doing some study on my own). I have tried several things (which I'll indicate below), but I cannot seem to figure it out. Here is how the problem is presented:
Problem: "Let $f$ be a continuous real-valued function on $[a,b]$. Suppose there exists a constant $M \geq 0 $ such that
$$|f(x)| \leq M \int_a^x |f(t)| dt$$
for all $x \in [a,b]$. Show that $f(x)=0$ for all $x \in [a,b]$."
My Thoughts: I have tried using the mean value theorem iteratively, but that seems to always lead me down a dead end road. I deduced that $f(a)=0$. If only $f$ were assumed to be differentiable, then maybe I could play with trying to get the derivative to be $0$, but unfortunately it's only continuous. My other thought was to (somehow) use the condition to show that $\int_a^b |f(x)| dx = 0$. I also played around a bit with contradiction, but to no avail. Even if one of these hair-brained thoughts is correct, I am not really sure what to do next.
If you have any ideas, suggestions, or solutions, I would really appreciate it if you are willing to share them. Thank you for your time.