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

 

[Stephen Tashi]

does it mean that it is continuous in both?

By "is continuous in both", do you just mean "is continuous"? Then no.. See Ch 9, section 1 of Counterexamples In Analysis (p 115 of the book, p 140 of the PDF) http://www.kryakin.org/am2/_Olmsted.pdf