Does continuity on rays imply measurability? Let us define a function on $\mathbb{R}^n$ by defining it on all lines crossing the origin $y=tx$,  $t\in\mathbb{R}$. It is defined to be continuous on these lines. Is it known to be measurable?
Let A be non-mesurable subset of $\left\lbrace x\in \mathbb{R}^n : \|x\| = 1 \right\rbrace$,

Then $B = \left\lbrace tx\in \mathbb{R}^n : x\in A ,  t\in\mathbb{R} \right\rbrace$ 

And now define the function f as :

$f(x) = 0$ if $x \not\in B$ and $f(x) = {\|x\|}$ if $x\in B$

then $f^{-1}(\left\lbrace 1 \right\rbrace) = A$, that is non measurable
Let A be non-mesurable subset of $\left\lbrace x\in \mathbb{R}^n : \|x\| = 1 \right\rbrace$,

Then $B = \left\lbrace tx\in \mathbb{R}^n : x\in A ,  t\in\mathbb{R} \right\rbrace$ 

And now define the function f as :

$f(x) = 0$ if $x \not\in B$ and $f(x) = {\|x\|}$ if $x\in B$

then $f^{-1}(\left\lbrace 1 \right\rbrace) = A$, that is non measurable