Continuity of a Hölder continuous function depending on a temporal parameter

Question

Let

• $(M,d)$ be a metric space
• $E$ be a normed $\mathbb R$-vector space

Now, let $$\left\|f\right\|_{\tilde C^{0+\alpha}(A,\:E)}:=\sup_{\stackrel{x,\:y,\:x',\:y'\:\in\:A}{x\:\ne\:x',\:y\:\ne\:y'}}\frac{\left\|f(x,y)-f(x',y)-f(x,y')+f(x',y')\right\|_E}{{d(x,x')}^\alpha{d(y,y')}^\alpha}$$ for $f:M\times M\to E$ and $A\subseteq M$ and $$\tilde C^{0+\alpha}(M,E):=\left\{f:M\times M\to E\mid\left\|f\right\|_{\tilde C^{0+\alpha}(K,\:E)}<\infty\text{ for all compact }K\subseteq M\right\}$$ be equipped with the topology induced by $$\left\{\left\|\;\cdot\;\right\|_{\tilde C^{0+\alpha}(K,\:E)}:K\subseteq M\text{ is compact}\right\}.$$

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Let $f:[0,\infty)\times M\times M\to\mathbb R$. Assume

1. $f(t,\;\cdot\;,\;\cdot\;)\in\tilde C^{0+\alpha}(M,\mathbb R)$ for all $t\ge0$
2. $f(\;\cdot\;,x,y)$ is continuous and nondecreasing for all $x,y\in M$

Are we able to deduce that $f$ considered as a mapping $[0,\infty)\to\tilde C^{0+\alpha}(M,\mathbb R)$ is continuous?

Answer 1

No, let $(M,d)$ be $[0,1]$ with the standard metric, let $E$ be $\mathbb{R}$ with the standard norm, and let $\alpha = 1$. Note that since $[0,1]$ is compact the topology on $\tilde C^{0+1}([0,1],\:\mathbb{R})$ is actually given by that semi-norm.

I'm going to describe a non-decreasing sequence of elements of $\tilde C^{0+1}([0,1],\:\mathbb{R})$ and then $f$ will be given by a linear interpolation between them. For $n=0,1,2,\dots$, let $g_n(x,y)=1-2^{-n-2}\left(3 + \cos(2^{n+1}\pi(x+y)) \right)$.

Note three things:

• $g_n(x,y)\leq g_{n+1}(x,y)$ for all $n$, $x$, and $y$.
• $\left\|g_n\right\|_{C^{0+1}([0,1],\:\mathbb{R})} \geq \frac{2^{-n-2}|\cos(2^{n+1}\pi(2^{-n-1}+2^{-n-1})) - 2\cos(2^{n+1}\pi (2^{-n-1}+0))+\cos(0))|}{|2^{-n-1}-0||2^{-n-1}-0|}=2^{n+2}$, by letting $x,y=2^{-n-1}$ and $x^\prime , y^\prime =0$.
• $g_n(x,y)\rightarrow 1$ pointwise as $n\rightarrow \infty$.

Now choose $f(t,x,y)$ so that $f(1-2^{-n},x,y)=g_n(x,y)$ for all $n$, $x$, and $y$; $f(t,x,y)=1$ for all $t\geq1$; and let $f(t,x,y)$ for $1-2^{-n}<t<1-2^{-n-1}$ be $\frac{(1-2^{-n-1})-t}{2^{-n-1}}g_n(x,y)+\frac{t-(1-2^{-n})}{2^{-n-1}}g_{n+1}(x,y)$ (i.e. the linear interpolation between $g_n$ and $g_{n+1}$.

Now $f$ satisfies the assumptions: $f(t,\cdot,\cdot)\in C^{0+1}([0,1],\:\mathbb{R})$ for every $t$ and $f(\cdot,x,y)$ is continuous and non-decreasing for every $x$ and $y$, but $f(t,\cdot,\cdot)$ is not continuous at $t=1$, since $\left\|f(t,\cdot,\cdot) \right\|_{C^{0+1}([0,1],\:\mathbb{R})} = 0$ when $t=1$, but it is unbounded as $t\rightarrow 1$ from below.