Consider $\mathcal C^0$ = {$f : (0, 1) → \mathbb R$ | f is continuous} and $E_{\beta}(f) = f(\beta)$, for $\beta \in (0,1)$, and the metric $$ d(f,g) = ||f-g||_2 = \sqrt{\int_0^1(f(x) - g(x))^2dx}$$.
I want to check whether $E_{\beta}$ is discontinuous for all $\beta$. This would mean, I need $d(f,g) < \delta$ (that is, the area between f and g to be arbitrarily low) but $d(f(\beta),g(\beta))$ to be equal to some constant, say 1.
Is it possible for two functions f and g to have a 'small' area between their graphs, and still satisfy something like, say $f(\beta) = g(\beta) + 1$ for all $\beta \in (0,1)$?