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The depth of a lake at a center point is a function of the distance of that point from shore.
Yes - if the domain is a single point. Not much point in having such a function, but it can exist.
Infinity, by definition, is not a point. Infinite means unending, so it cannot be located at a point. The moment at which a function reaches infinity is when that function ceases to be bounded.
The graph of the sine function is periodic at every point. Periodic means that the value of the function at every point is repeated after an integer multiple of the period.
If the second derivative of a function is zero, then the function has a constant slope, and that function is linear. Therefore, any point that belongs to that function lies on a line.