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2006-05-30, 18:26   #4
ewmayer
2ω=0

Sep 2002
República de California

25×307 Posts

Quote:
 Originally Posted by Damian Suppose the function: f(x,y) ={ 1 (if x=0 or y=0) { 0 (otherwise) the partial derivatives at the origin are df/dx = 0 and df/dy = 0 so they exist and are continious. However the function is not continious in (0,0), so it can't be differentiable there. (there can't be a tangent plane) What did I misundertood? Because I think the theorem says that if the partial derivatives exists and they are continious then the function is differentiable there.
That function is not differentiable at the origin - just use the definition of derivative in the sense of the limit of the divided difference delta(f)/delta(coordinate) - that limit does not exist at (0,0) in this case.

Also, it makes no sense to speak of a function being continuous at a single point - continuity only makes sense in regions, i.e. neighborhoods of a point.