Question

Rd if and only if f1(F) is close Show that a function f : Rd -> Rm is continuous on for each close set F in R"

Please show all steps. thank you

Answer 1

Solution :

First let's assume f is continuous.

Then let F be a closed set in R^m,then F^c will be open.

=> by definition, f^-1(F^c) is open as f is continuous.

Now f^-1(F^c) = {f^-1(F)}^c is open => f^-1(F) is closed.

Conversely let f^-1(F) is closed for every closed set F in R^m.

Now let G be an open set in R^m.then G^c is closed,so f^-1(G^c) = {f^-1(G)}^c is closed,so f^-1(G) is open.hence f is continuous

Hence proved.