**Well-known example:** Consider the function 
$$f(x,y)=\left\{\begin{array}{c}
\frac{x^2y}{x^4+y^2} & \text{if }(x,y)\neq(0,0)\\
0 & \text{if }(x,y)=(0,0)
\end{array}\right.$$
When restricted to any straight line through the origin, this function is continuous. However, if we approach the origin along the parabola $y=x^2$, we get a limit of $\frac 12$, so $f$ is actually discontinuous. The question is whether smooth curves can always ferret out discontinuity in this way.



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> Does there exist a function $f:\mathbb R^2\to \mathbb R$ which is discontinuous at a point $x$, but is continuous at $x$ when restricted to any smooth curve?
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Discontinuity is witnessed by a sequence $\{x\_i\}$ converging to $x$ so that $\{f(x\_i)\}$ does not converge to $f(x)$. Since we could take $f$ to be the characteristic function on $\{x\_i\}$, the above question is equivalent to the following.



> 
> Given a convergent sequence $\{x\_i\}$ in $\mathbb R^2$, must there be a smooth curve passing through infinitely many of the $x\_i$?
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Can continuity of a function be checked by restricting to smooth curves?
Usually, by a "smooth curve through $0\in\mathbb R^2$", one means a $C^k$ mapping $f:[-a,a]\to\mathbb R^2$ where $k\ge 1$ is the desired degree of smoothness, $a>0$, $f(0)=0$, and $f'(x)\ne 0$ for all $x\in[-a,a]$ (without the last condition, you can get quite ugly images even of $C^\infty$ mappings). Equivalently, a smooth curve is a graph of a $C^k$-function near the origin after some rotation of coordinates. Now, the answer depends on $k$. If $k=1$, then there is a subsequence $x\_n$ approaching $0$ from some limiting direction. Choosing this direction as the positive semiaxis, we see that we can rarefy the sequence a bit and put everything on a $C^1$ graph with $0$ derivative at the origin. On the other hand, to request any particular modulus of continuity for $f'$ is already impossible because that would require some estimate of the kind $|\mbox{arg}(x)|\le \omega(|x|)$ with some fixed function $\omega$ tending to $0$ at $0$ where $\mbox{arg}$ is measured from the limiting direction. However, we can construct a sequence of points whose absolute values tend to $0$ very fast while their arguments tend to $0$ very slowly. 


