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Finding an upper bound on composite $C^{1}$ functions

The conditions for the problem I am currently tackling is as follows: Let $T, M > 0$. Define : $X_{T} := \{ u \in C^{1} ([0,T]) : ||u||_{T} \leq 2M \}$ where $||u||_{T} \ = \ ^{\text{sup}}_{t \in (...
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