Proving partial deviatives not continous

[wowowo2006]

Homework Statement

f(x,y) = y^2 + (x^3)*sin(1/x) when x =/= 0
= y^2 when x = 0

i want to prove fx(x,y) is not continuous at (0,0)

Homework Equations

The Attempt at a Solution

i found when x=/=0 , fx = 3(x^2)sin(1/x) - xcos(1/x) -----eq(1)
and limit(x,y -> 0,0) eq(1) = 0 as sin and cos is bounded
and the actual fx(0,0) = limit(h->0) (f(h,0)-f(0,0))/h = lim(h->0) (h^2)*sin(1/h) = 0
it seem limfx(0,0) = fx(0,0)
so i cannot conclude that fx is not continuous at (0,0)
where did i go wrong?

 

[wowowo2006]

can someone offer help please?.

 

[Millennial]

I think the partial derivative is actually continious. Graphing it shows it has no oscillation, which is a quantitative definition of continuity.