Let
- $T>0$
- $I:=(0,T]$
- $d\in\mathbb N$
- $K\subseteq\mathbb R^d$ be compact
- $f:\overline I\times K\times K\to\mathbb R$ be (jointly) continuous
- $\beta\in(0,1]$
Moreover, let $$\left\|g\right\|_{C^{0+\beta}(K)}:=\sup_{x,\:y\:\in\:K}\left|g(x,y)\right|+\sup_{\stackrel{x,\:y,\:x',\:y'\:\in\:K}{x\:\ne\:x',\:y\:\ne\:y'}}\frac{\left|g(x,y)-g(x,y')-g(x',y)+g(x',y')\right|}{|x-x'|^\beta\left|y-y'\right|^\beta}$$ for $g:K\times K\to\mathbb R$ and $$C^{0+\beta}(K):=\left\{g:K\times K\to\mathbb R\mid\left\|g\right\|_{C^{0+\beta}(K)}<\infty\right\}.$$ Assume $$f(t,\;\cdot\;,\;\cdot\;)\in C^{0+\beta}(K)\;\;\;\text{for all }t\in\overline I\tag1.$$
Are we able to show that $$\overline I\ni t\mapsto\left\|f(t,\;\cdot\;,\;\cdot\;)\right\|_{C^{0+\beta}(K)}\tag2$$ is continuous?
In general, if $E_i$ is a compact metric space and $h:E_1\times E_2\to\mathbb R$ is (jointly) continuous, then $$E_2\ni x_2\mapsto\sup_{x_1\:\in\:X_1}f(x_1,x_2)\tag3$$ is continuous (the crucial argument is the uniform continuity of $h$ implied by the compactness of $E_1\times E_2$).
So, the dependence on $t$ of the first term in the definition of $\left\|f(t,\;\cdot\;,\;\cdot\;)\right\|_{C^{0+\beta}(K)}$ is continuous. What's about the second term?