TITLE: Continuous on every continuous path vs Continuous
QUESTION [5 upvotes]: Suppose $F:\mathbb{R}^2\rightarrow\mathbb{R}$ is such that for all continuous paths $g:[0,1]\rightarrow\mathbb{R}^2$, $F(g(t))$ is continuous.
Is F continuous?

REPLY [2 votes]: Yes:  It suffices to show that $F(x_k)$ converges to $F(x)$ for any sequence $(x_k)$ in $\mathbb R^2$ that converges to $x$.  This convergence follows form the consideration of $g$ whose image is the polygonal path along this sequence ($g(1-1/k) = x_k$, $g(1) = x$).