The last term being a function of c(t) seems like rather a big problem with the proof. So far as I know, there isn’t any requirements on c other than it be monotone, so the function f^n(c(t)) could be non-continuous even if the derivative is continuous.
I don’t see how the MVT for integrals gets you out of this, but I’ll think about it some more, since I agree it’d be a nicer proof than the standard one if it can be fixed.

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