Answer
False
Work Step by Step
The given information is only valid for continuity. However:
\[
\lim _{\Delta x, \Delta y \rightarrow(0,0)} \frac{f(x+\Delta x, y+\Delta y)-f\left(x_{0}, y_{0}\right)}{\sqrt{(\Delta y)^{2}+(\Delta x)^{2}}}=0
\]
Which can be approximated by:
\[
\lim _{\Delta x . \Delta y \rightarrow(0,0)} \frac{\Delta f-f_{x}\left(x_{0}, y_{0}\right) \Delta x-f_{y}\left(x_{0}, y_{0}\right) \Delta y}{\sqrt{(\Delta y)^{2}+(\Delta x)^{2}}}=0
\]
Therefore, the above statement is false.
