Knowlpedia

Continuity via sequences

In metric spaces, f is continuous at x iff it preserves limits of sequences converging to x

Continuity via sequences: Let (X,dX)(X,d_X) and (Y,dY)(Y,d_Y) be and let f:X→Yf:X\to Y. Then ff is x∈Xx\in X if and only if for every sequence (xn)(x_n) in XX, xn→x⟹f(xn)→f(x).x_n\to x \quad\Longrightarrow\quad f(x_n)\to f(x).

Remarks

This is one of the most-used characterizations of continuity in analysis: it converts an ε\varepsilon–δ\delta condition into a limit-preservation property.