Meaning of functions that vanish at a point

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.

Solution 1:

Vanish just means the function is $0$.

Solution 2:

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