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