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On the page 32 of Lee's book Manifolds and differential geometry, he writes:

In the definition of path connectedness..., we used continuous paths, but it is not hard to show that if two points on a smooth manifold can be connected by a continuous path, then they can be connected by a smooth path.

I do not quite understand.

First of all, what does a path being smooth mean? A path on a manifold is a function: $$ p:[0,1]\rightarrow M $$

I think to be continuous is in the topological sense.

But what does smooth mean? Is it in the diffeomorphic sense?

Do we regard $[0,1]$ as a set in the manifold $\mathbb{R}$ and applying the definition of smoothness of function between manifolds?

After all, I wonder how to prove this statement.

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In down-to-earth terms it means that the derivative $p^\prime(t)$ exists at every $t\in(0,1)$. Then, the claim is that every continouous path in a smooth manifold can be deformed to a smooth map. This results from a Smoothing Lemma which is usually given for granted in any graduate course on the subject ... :) – Andrea Mori Oct 22 '12 at 11:15
..all right.. I will take it for granted too..:) – hxhxhx88 Oct 22 '12 at 12:04

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