difficulty in proving ''If f is continuous then $\left\|{f}\right\|=0$ implies $f(x)\equiv 0$

Question

I have read in a textbook that '' if a function f(x) is continuous then $$\left\|{f}\right\|=\sqrt{\left({f(x),f(x)}\right)}=\sqrt{\int\limits_{a}^{b}{r(x)f(x)^{2}dx}}=0$$ implies that $$f(x)\equiv 0$$ (The weight function r is positive) How is it possible that a non-zero discontinuous function has zero norm?