Continuity of $\delta$ in the definition of continuity

The function $\delta$ may be defined as follows. For every $x\geqslant 0$, let $F(x)=\sup\{|f(z)|\,;|z|\leqslant x\}$. Then $\delta(0)=0$ and, for every $t\gt0$, $\delta(t)=\sup\{x\gt0\,;\forall z\lt x, F(z)\lt t\}$. One sees that the question really concerns the nondecreasing function $F$ defined on $[0,+\infty)$.

The function $\delta:t\mapsto\delta(t)$ is defined on $[0,+\infty)$ and continuous on the left on $(0,+\infty)$. Let $t\geqslant0$. The function $\delta$ is continuous on the right at $t$ if and only if $F$ is strictly increasing on the right at $t$, that is, if and only if $F(s)\gt F(t)$ for every $s\gt t$.

Now, $F$ is always nondecreasing and $F$ is increasing if and only if the function $g$ defined on $[0,+\infty)$ by $g(x)=\max\{|f(x)|,|f(-x)|\}$ for every $x\geqslant 0$, is increasing.

The corresponding necessary and sufficient condition in the general case is that $g$ is increasing where, for every $x\geqslant0$, $g(x)=\sup\{|f(z)|\,;\|z\|=x\}$.