Question

If a function f(x,y) is not continuous at the point (a,b), will it be possible that \(\frac{\partial}{\partial x}\)f(x,y) is continous at (a,b)?

Answer 1

consider \[f(x,y)=x+\frac{1}{y}\] and the point \((1,0)\)

Answer 2

\(f_x = 1\) ?

Answer 3

is \(f_x=1\) continuous at (1,0)?

Answer 4

Seems to be continuous at (1,0)