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Let $f:(a,b)\to \mathbb{R}$ be a continuous function and define $\Gamma(f) = \{(x,f(x)):x\in (a,b)\}$. The two maps $\Psi: \Gamma(f)\to (a,b)$ given by $(x,f(x))\mapsto x$ and $\Phi: (a,b)\to \Gamma(f)$ given by $x\mapsto (x,f(x))$ are both continuous and inverse to each other. Therefore $\Gamma(f)$ is homeomorphic to $(a,b)$, and $\{(\Gamma(f),\Psi)\}$ is an atlas consisting of a single chart. This means $\Gamma(f)$ is a smooth manifold.

This confuses me greatly. Somehow I feel $\Gamma(f)$ should be a smooth manifold only when $f$ is smooth, not merely continuous. What is wrong here?

mez
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1 Answers1

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It is true that $\Gamma(f)$ can be given the structure of a smooth manifold, since as you note it is homeomorphic to an interval. The sense in which it is not smooth if $f$ is not smooth is that it is not a smooth submanifold of $\mathbf{R}^2$, since the chart you have defined composed with the inclusion map (giving a map $(a,b)\to \mathbf{R}^2$) will not be smooth.

ajd
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