For a one-dimensional real valued function in the usual topology. It is said that f(x) is continuous in x0if there exist limit at x0 and its valut is f(x0)

∀ ε >0, ∃ δ >0 : si |x - x0| < δ => | f(x) - f(x0) | < ε

Note that for f is continuous on x0, f must first be defined in x0 (which is not necessary in limit definition).