Problem 5-6 in Michael Spivak's Calculus on Manifolds reads:
If $f:\mathbb R^n\to\mathbb R^m$, the graph of $f$ is $\{(x,y):y=f(x)\}$. Show that the graph of $f$ is an $n$-dimensional manifold if and only if $f$ is differentiable.
(here manifold means really submanifold, &c, and generally the statement has to be read in the context of the book, of course)
Now, this statement suffers from counterexamples: for example, the graph of the function $f:t\in\mathbb R\mapsto t^{1/3}\in\mathbb R$ is a submanifold.
I only have access to a printing of the original edition of the book: anyone happens to know if newer editions have the statement changed?
I can prove the statement if I change it to read «the graph $\Gamma$ of $f$ is an $n$-dimensional manifold and the differential of map $\Gamma\to\mathbb R^n$ given by projection on the first $n$ components has maximal rank everywhere if and only if &c». This seems like a rather strong hypothesis: can you think of a weaker one which will still give a sensible true statement? (I don't like this hypothesis because at that point in the book one does nto yet have the differential available)
Later. MathSciNet tells me there is a russian translation. Maybe one of our Russian-enabled friends on the site can tell me if the problem is there too? Translation into Russian traditionally includes some fixing, irrc :)