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Sep 5, 2014 at 14:35 comment added Pietro Majer Note that the continuous dependence from $x$ is neither sufficient nor necessary for the positive answer to hold. If $\{f(\cdot, x)\}_x$ is an equicontinuous family of functions on $[0,1]$, then either $C(t)=\infty$ for all $t$ or $C$ is a continuous real valued function of $t$ (thus bounded).
Sep 5, 2014 at 14:22 vote accept Benji
Sep 5, 2014 at 14:20 history answered Pietro Majer CC BY-SA 3.0