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Consider the graphs of the functions $f_1(x) = |x|$, and $f_2(x) = x$ under the subspace topology of $\mathbb{R}^2$.

Both of these graphs are smooth manifolds, just pick coordinate charts to be $(x, f_i(x)) \leftrightarrow x$.

Moreover, they are diffeomorphic via the map $(x, f_1(x)) \rightarrow (x, f_2(x))$.

This seems to clash with my intuition. For example, the graph of $f_1$ has a corner, so it "shouldn't" be smooth, much less diffeomorphic to $f_2$, which is just a straight line.

Can someone explain what's going on here?

In light of these examples, how should I visualize smooth manifolds and diffeomorphisms?

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    There's [an old question](http://math.stackexchange.com/questions/45673/is-m-x-x-x-in-1-1-not-a-differentiable-manifold) on the first manifold. I think Prof Wong's answer could help a lot, here.2011-09-04
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    I think maybe your intuition isn't fully thought-out. You're giving a set a smooth structure based on a bijection with $\mathbb R$ -- this is not a very natural thing to do. You can make the Cantor set a smooth manifold diffeomorphic to $S^n$ or $\mathbb R^n$ for any $n \geq 2$ using this technique, so it's not particularly interesting.2011-09-04

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