# Continuity and differentiability in two variables

wavingerwin
Hi

If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e.
$f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2)$ and $f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2)$
for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##,

does it mean that it is continuous in both?
$f(x+d_x,y+d_y) = f(x,y) + \Delta_xd_x +\Delta_yd_y+O(d_x^2,d_y^2)$

How about differentiability? (if the function is independently differentiable in ##x## and ##y##, is it differentiable in both ##x## and ##y##?)

Cheers
wavingerwin