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Old 2006-05-30, 18:26   #4
ewmayer's Avatar
Sep 2002
Rep├║blica de California

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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.
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