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Let $g:[0,T]\times \Omega \to \mathbb{R}$ where $\Omega$ is spatial domain, and $f:\mathbb{R} \to \mathbb{R}$.

Is it possible for $$\frac{d}{dt}f(g(x,t))=f'(g(x,t))g'(x,t)$$ to be continuous in $t$, but for $g'$ to be discontinuous in $t$?

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Yes, the simplest case is when $f$ is a constant function. A less simple case would be that $f'$ = 0 whenever $g'$ is discontinuous. Then (at least for jump discontinuities) the RHS should tend to zero from either direction.

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