Let $f:=f(x,t)$ belong to the parabolic Hölder space $C^{1,\beta}(U\times [0,T])$ where $U$ is a compact subset of $\mathbb R^2$ and $0<\beta <1$. This yields that $f$ is a Hölder continuous function in both space and time variables with exponents $1$ and $\beta$ correspondigly.
If not mistaken, the Hölder continuity of $f$ implies that $f$ is also uniform continuous in both variables.
QUESTION: If I consider $f$ as a function only of time $t$ then is $f(\cdot,t)$ uniformly continuous in the sense that $\; \exists \delta >0$ with $\vert t_1-t_2 \vert \le \delta$ such that $\vert f(\cdot,t_1)-f(\cdot,t_2) \vert < \epsilon$ for some $\epsilon >0$ $(*)$?
If my whole argument is false, then is the Hölder continuity sufficient in order to deduce something similar to $(*)$ for $t \to f(\cdot,t)$?
The more I'm getting my head around this, the more puzzled I get so I would appreciate any help or hint.
Thanks a lot in advance!