- #1
athrun200
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We have a corollary that
But I wonder can we prove a function is not differentiable by showing that [itex]f_{x}[/itex] or [itex]f_{y}[/itex] are not continuous?
i.e. is the converse of this statement true?
By the way, are there any books have a proof on this corollary?
Most of the Calculus book state the corollary of theorm only without prove.
"If the partial derivatives [itex]f_{x}[/itex] and [itex]f_{y}[/itex] of a function f(x,y) are continuous throughtout an open region R, then f is differentiable at every point of R."(Thomas F. 1994. Calculus. U.S.A. Wesley p824)
But I wonder can we prove a function is not differentiable by showing that [itex]f_{x}[/itex] or [itex]f_{y}[/itex] are not continuous?
i.e. is the converse of this statement true?
By the way, are there any books have a proof on this corollary?
Most of the Calculus book state the corollary of theorm only without prove.