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A non-differentiable function with all directional derivatives

Define f(x,y) to be tex2html_wrap_inline609 . Then for any unit vector tex2html_wrap_inline611 , the directional derivative at the origin is by definition

displaymath613

In particular, tex2html_wrap_inline615 and tex2html_wrap_inline617 (since these are directional derivatives for tex2html_wrap_inline619 and tex2html_wrap_inline621 respectively). If f were differentiable at the origin, then tex2html_wrap_inline625 would equal tex2html_wrap_inline627 for every tex2html_wrap_inline629 . But tex2html_wrap_inline631 is not always zero, so f is not differentiable.

Here's the graph of f:

external


Tom Vogel
Mon May 5 12:53:33 CDT 1997