I have a Banach space $X$ and a linear functional $f:X\rightarrow \mathbb{C}$.
The oscillation of $f$ at a point $x$ is defined as $$\omega_f(x) : \lim_{\delta \rightarrow 0^+} [ \sup_{y,z \in B_\delta(x)}|f(y)-f(z)| \ ]$$ My question is: can we prove that the function $\omega_f(x): X \rightarrow \overline{\mathbb{R}}_+$ is continuous? Thanks
Let $x,x'\in X$. Then since $f$ is linear, the values it takes on $B_\delta(x')$ are just the same as the values it takes on $B_\delta(x)$ but translated by $f(x')-f(x)$. It follows immediately that $\omega_f(x)=\omega_f(x')$, so $\omega_f$ is constant.
(In fact, as mentioned in the comments, any discontinuous linear functional is unbounded so it takes unbounded values on any ball, so $\omega_f$ is always $\infty$ unless $f$ is continuous so it is always $0$.)