Hi dear friends here, I got a question, it supposed to be quite simple, but it is a little bit tricky, can anyone please help me? Thanks a lot.

Here it is:

f(x,y)={x^3/(x^6+y^2), for (x,y) does not equal (0,0)}

f(x,y)={0, for (x,y)=(o,o)}

Show that df/dx exists at (0,0) but is not continuous there.

I used first principle to prove the limit of is 0 at (x,y)=(0,0), which means it does exist, am I right? What is the exact definition to say something exists?

Then, I can imagine it is discontinuous because at (0,0) the graph of should be broken, which has the circle there, but I just forgot how to show it. I thought the definition of "continuiy" is the value of limit equals to the value of function. So how do I write about it? Please help me, thanks a lot.