- #1
Felafel
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- 0
Homework Statement
I have the function f, defined as follows:
f=0 if xy=0
f= ##xysin(\frac{1}{xy})## if ##xy \neq 0##
Study the differentiability of this function.
The Attempt at a Solution
there are no problems in differentiating the function where ##xy\neq0##.
the partials in (0,0) are both zero.
If i consider xy as a single variable i get from the definition of differential that:
##\lim_{xy \to 0} \frac{f(xy)-f(0)-\nabla f(0)(xy)}{\sqrt{(xy)^2-0}}##=##\lim_{xy \to 0} \frac{xy sin(1/xy)}{xy}## doesn't exist
On the other hand, if i consider x,y separately i get:
##\lim_{(x,y) \to (0,0)} \frac{f(x,y)-f(0,0)-\nabla f(0,0)(x,y)}{\sqrt{x^2+y^2}}##=##\lim_{xy \to 0} \frac{xy sin(1/xy)}{\sqrt{x^2+y^2}}=0## and thus the function is differentiable even when x,y are 0.
these two results are conflicting. what am i doing wrong?