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docnet

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- Homework Statement
- Assume that if the real-valued function ##h(x)## is Lipschitz continuous, then it can be proven the function ##g(h(x))## is also Lipschitz continuous. If ##f(x)## is given to be only continuous and not Lipschitz, does the above result say anything about the continuity of ##g(f(x))##?

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I think the answer is no, since the requirements for Lipschitz continuous and epsilon-delta continuous are different.

The reason I'm asking such an odd question is, I made a mistake by writing a proof of the Lipschitz continuity of ##g(h(x))## using the assumption that ##h(x)## is Lipschitz continuous. Because, it turned out I was supposed to use ##f(x)## epsilon-delta continuous. It got me thinking.. would it be possible to get the continuity of ##g(f(x))## using the completed proof about Lipschitz continuity?

The reason I'm asking such an odd question is, I made a mistake by writing a proof of the Lipschitz continuity of ##g(h(x))## using the assumption that ##h(x)## is Lipschitz continuous. Because, it turned out I was supposed to use ##f(x)## epsilon-delta continuous. It got me thinking.. would it be possible to get the continuity of ##g(f(x))## using the completed proof about Lipschitz continuity?

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