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I realize this may be a very thick question, but I have been wondering for some time. Sometimes I am asked to prove or read proofs involving "functions that vanish at a point" or "every point" or something along these lines. The problem is I do not know what it means for a function to vanish at a point. It sounds like it means the function goes to 0 as a sequence of points arrives at the point, but if the function is continuous, doesn't that just mean the function is equivalently 0 at the point? I am basically confused what "vanish" means.

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It simply means that the function is $0$ at the point in question. – Brian M. Scott Oct 6 '12 at 1:14
@BrianM.Scott So, I am reading a proof of the fact that "a function of x,y in a region whose partial derivatives vanish at every point in that region is constant". Isn't this trivial, since that just implies the function's partial derivatives are 0, and thus the function must be constant? – tacos_tacos_tacos Oct 6 '12 at 1:16
It’s not quite trivial: the partial derivatives are actually directional derivatives, so the fact that they vanish immediately says only that the function is constant in two specific directions. It takes a little work to get from that to the function’s actually being constant. – Brian M. Scott Oct 6 '12 at 1:19
Why is this trivial? – Marra Oct 6 '12 at 1:19
Face palm. Oops. Right, u_x + u_y != du/dz. Sorry, it was thicker than I thought. – tacos_tacos_tacos Oct 6 '12 at 1:24
up vote 3 down vote accepted

Vanish just means the function is $0$.

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Customarily, it means the function has a zero at the point or on the set. I usually just say the function is zero there.

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Yeah it basically means that the function outputs the value zero at that point. I guess this can be interpreted as a "thick" answer. Also in case of a continuous function on a closed interval which can be looked at as a sequence of uniformly converging polynomials, this might mean something more. Maybe someone can comment on this, if there is anything to comment. It just occurred to me and might not mean much.

Also, this is not directly tied into your question, but there's what is called the support of a function. It is those set of points in the domain at which the function has a non-zero value or the closure of this set. If this set is compact, then it leads to several interesting results in analysis.

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