Let ${\cal C}[0,1]$ be the space of real valued continuous function on $[0, 1]$. Fix $\psi \in {\cal C}[0,1]$ and define $$\rho_{\psi}(f, g):= \int_0^1 \psi(x) |f(x) - g(x)| dx, \;\; f, g, \in {\cal C}[0,1].$$
Show that if $\psi(x) = \left\{ \begin{array}{ll} 0, & 0\leq x \leq \frac{1}{2} \\ x - \frac{1}{2}, & \frac{1}{2} < x \leq 1\\ \end{array}\right.$ for all $x \in [0,1]$, then $\rho_{\psi}$ is a not a metric on ${\cal C}[0,1]$.
I am stuck in this problem. It seems the Positive Definite and Symmetric properties are okay. I was trying to find a counterexample to the triangle inequality but to no avail. For instance, I was trying $f(x)=x^2$, $g(x)=x$ and $h(x)=x^3$.
If $f$ and $g$ are functions that agree on $(1/2, 1]$, then $\rho_{\psi}(f, g)=0$ even if $f$ and $g$ are distinct.