How function is continuous in 2 or 3 variable?

Answer 1

In the same way that a fuction is continuous is one varieable, except the neighbourhood of the arguments and the function value are defined using multi-dimensional metrics. Usually these are the multi-dimensional equivalents of the Pythagorean metric.

Thus, for 2-d

a function f(x,y) is said to be continuous at the point (a,b) if, given e > 0, however small, you can find a value d such that

|f(x,y) = f(a,b)| < e for all (x,y) such that |(x,y) - (a,b)| < d

In other words, you can find a value d such that for ALL points (x,y) that are near enough to (a,b), you can make f(x,y) arbitrarily close to f(a,b).

The 3 (or more) variable definition is analogous.