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I was reading a bit in "Introduction to Smooth Manifolds" by John M. Lee for some motivations and came across the following section on page $11$-$12$. "Each point in $M$ is in the domain of a coordinate map $\varphi:U \to \hat{U}.$ A plausible definition of a smooth function on $M$ would be to say that $f:M \to \mathbb{R}$ is smooth if and only if the composition $f \circ \varphi^{-1}:\hat{U} \to \mathbb{R}$ is smooth in the sense of ordinary calculus. But this will only make sense if this property is independent of the choice of coordinate chart."

Why is that the case? Wouldn't the definition "$f$ is continuous iff at every point $p$ there is some chart $(U,\varphi)$ with $p \in U$ such that $f \circ \varphi^{-1}$ is smooth" be a perfectly fine definition? Maybe I am overlooking something basic here, but I, at the moment, don't see why this wouldn't be well defined.

user3118
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    Suppose you have two functions f and g, and that both are smooth at p according to your definition, but you need to use different charts for each. Will f+g be smooth at p? – Mariano Suárez-Álvarez Jan 23 '23 at 21:31
  • @MarianoSuárez-Álvarez Well in general I couldn't claim that. My question was two-fold though, I wanted to know whether this would be a fine definition to make in the sense of being well defined and secondly I wanted to know why this definition wouldn't make sense. As I said, this should be well defined so only the second question has to be answered. Your comment suggests that John M. Lee meant that this definition isn't suitable if one wants to have expected properties of smooth functions, right? In that context I agree and see what he meant now. – user3118 Jan 23 '23 at 21:39

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