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Consider $\phi: A\times B \to C$, with all spaces involved topological spaces. $\phi$ is continuous if for any given neighborhood of the image point, $N_{\phi(a,b)}$, there exist neighborhoods $N_a$ and $N_b$ in the domain, such that $\phi(N_a,N_b)\subset N_{\phi(a,b)}$.

This is a pretty clear notation, it tells us that continuity of $\phi$, first of all, guarantees the existence of the neighborhoods $N_a$ and $N_b$ and that it maps those neighborhoods inside a given neighborhood of the image. This is all continuity has to offer.

If we also required smoothness, not mere continuity of $\phi$, what would the condition be on neighborhoods? In other words, are we led naturally to the definition of manifolds? There is no metric to define derivatives on topological spaces. How do we do it consistently?

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    $\begingroup$ The thing is that smoothness is a property of linear approximation : it tells you that your function must locally look like something linear; and this "something" must in turn vary more or less smoothly, depending on the precise measure of smoothness you want to impose. But linearity is something we only know about for $\mathbb{R}$ or $\mathbb{C}$ (or, later, topological fields; e.g. $\mathbb{Q}_p$); so we have to have a resemblance to one of these to be able to speak of linear approximation $\endgroup$ – Max Feb 4 at 12:59
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First of all to talk about smoothness of $\phi$ we first have to ensure that we can give some smooth structures on given topological spaces i.e $A$, $B$, $C$ because there are topological manifolds with no smooth structures.Topological manifold but not "smooth" In fact for a topological space to be a smooth manifold first of all it has to be a topological manifold(Hausdorff, second countable and locally euclidean) and secondly it must have a smooth structure(existence of a smooth maximal atlas). In fact a same topological space can have many smooth structures on it. So to give a definite answer to your question first of all we need to know whether those topological spaces admits a smooth structures or not and if it admits then what structure it admits. I do not think the notion of smoothness will naturally follow from continuity.

There is an interesting result if topological spaces are Lie groups. It says any continuous homomorphism between 2 Lie groups is smooth.(Partially you can get some idea about naturally extending the concept of continuity to smoothness).Please check the link below. Continuous homomorphisms of Lie groups are smooth

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