Prove Integral Inequality: f Nonnegative, Continuous on [0,1]

[adpc]

Homework Statement

For f nonnegative and continuous on [0,1], prove.
$$\left( \int f \right) ^2 < \int f^2$$
With the limits from 0 to 1.

Homework Equations

The Attempt at a Solution

I was trying to use Upper sums, i.e.
$$\inf \sum \Delta x_i M_i(f^2) = \inf \sum \Delta x_i (M_i(f))^2$$
and then compare this to $$\inf \left[ \sum \Delta x_i M_i(f) \right] ^2$$
Am I in the correct way to prove it?
Why does f is required to be continuous, I didn't use this fact!

 

[micromass]

Yes, this is the correct way of showing this.

About the continuity of f. You don't really need that here. The inequality is good for any function. But you got to make sure that the integral exists. And that is probably why they chose f to be continuous, because otherwise the integral may not exist...