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My doubt is about to find the limit of a two variable function $ f ( x, y) $ at $(0,0)$. Let $$ \lim_{x\to 0} f (x, mx) =a \in \mathbb {R}$$

i.e. limit along any direction $ y=mx $ exist and is a fixed real number $a.$ Now can i conclude that $\lim_{(x, y)\to(0,0)} f (x, y)$ exists and equal to $ a? $ I am thinking the same because $ y=mx $ gives all direction. Am i right in this way? Please suggest me. Thanks in advance.

neelkanth
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    No, this is false. See here, for example: http://math.stackexchange.com/questions/93886/computing-a-multivariable-limit/93926#93926. – Hans Lundmark Oct 22 '15 at 12:50

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Another counterexample: Let $P=\{(x,x^2): x > 0\},$ and define $f$ to be $1$ on $P$ and $0$ everywhere else.

zhw.
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