TRUE OR FALSE: Suppose that f is a bounded function on [a, b]. If P is an arbitrary partition of interval [a,b] then U(f,P) ≤ L(f,P).

Question

TRUE OR FALSE: Suppose that f is a bounded function on [a, b]. If P is an arbitrary partition of interval [a,b] then U(f,P) ≤ L(f,P).

Answer 1

No

Here's a counter example:

f(x) = { 1 if x is rational,

0 if x is irrational

}