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Take a function $f:\mathbb{R}^2\to \mathbb{R}$. For any straight line $\ell\subset\mathbb{R}^2$, consider the restriction $f_\ell:\mathbb{R}\to\mathbb{R}$ of $f$ to $\ell$. I believe the following is true:

$f$ is continuous $\Leftrightarrow$ $f_\ell$ is continuous for every $\ell$.

I have no idea how to prove it, however. Moreover, I believe it to be true if we consider just horizontal ($\mathbb{R}\times\{0\}$) and vertical lines ($\{0\}\times\mathbb{R}$). Does somebody have a proof or a counterexample?

Daniel Robert-Nicoud
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    The interesting implication is false, see http://math.stackexchange.com/questions/174816/discontinuous-functions-that-are-continuous-on-every-line-in-bf-r2. The other one is trivial. – PhoemueX Mar 22 '15 at 17:41
  • isn't the domain of $f_\ell$ is $\ell\subset \Bbb R^2$? – user149418 Mar 22 '15 at 17:42
  • @user149418: Probably what is meant is that you choose an (arbitrary) parametrization of the line like $t \mapsto t \cdot (x_0, y_0) + m$, where $(x_0, y_0)$ is the dirction vector of the line and $m$ is some point on the line and then $f_\ell (t) = f(t \cdot (x_0, y_0) + m)$. – PhoemueX Mar 22 '15 at 18:30
  • @PhoemueX Thank you! Feel free to put your comment as an answer, and I will accept it. – Daniel Robert-Nicoud Mar 22 '15 at 23:40

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It is clear that continuity of $f$ implies that each function $f_\ell$ is continuous, either because restrictions of continuous functions are continuous or because compositions of continuous functions are continuous.

The reverse implication is false (i.e. it can happen that all functions $f_\ell$ are continuous, but $f$ itself is not). A nice example of this is given here: Discontinuous functions that are continuous on every line in $\bf R^2$.

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