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I knew if $f$ is Frechet differentiable at $x$ then $f$ is continuous at $x$.

My question is: Can we imply that $f$ is uniformly continuous or absolutely continuous if if $f$ is Frechet differentiable at $x$?.

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We have that $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=x^2$ is not absolutely continuous (not even uniformly continuous), yet $f$ is Fréchet differentiable with its Fréchet derivative given by the map $h \mapsto 2xh$.

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