I have the function $$d_1(h_\alpha,h_\beta)=\int_a^b|h_\alpha(t)-h_\beta(t)|dt,$$ where $RI[a,b]$ is the set of all Riemann Integrable functions, and $$h_\alpha=\{\alpha \text{ if }t=a \text{ or } t=b, 0 \text{ if } a<t<b\}.$$ I want to show that $d_1$ is not a metric on $RI[a,b]$.
I had the thought to prove this as the triangle inequality $d(h_\alpha,h_\beta)\le d(h_\alpha,h_\gamma)+d(h_\gamma,h_\beta)$ does not hold and this could be shown by counterexample.
From sketches, I can see that I could use $h_{-\alpha}(t)$ as this integral would leave $d_1=0$ but I can't see how to take this further into an actual written proof.
The triangle inequality holds in this context. But it turns out that we can have $d_1(f,g)=0$ even when $f\ne g$. Take, for instance, $f$ equal to the null function and$$g(x)=\begin{cases}0&\text{ if }x<b\\1&\text{ if }x=b.\end{cases}$$That's why $d_1$ is not a metric.