Hi Tryss

many thanks for your patience

I just wonder if lambda_0 is in [0,1] since that's required to have f(C)=0

because what I interpreted your answer was lambda_0 is outside of [0,1]

And the method you are using to prove continuity is: since {0}, {1} are open sets in {0,1} with discrete topology, we want to find open neighborhood(ball) B and C of x such that f(B)=0 if d(x,m)<lambda_{0}*d(m,C) and f(C)=1 if d(x,m)>lambda_{0}*d(m,C). Am I right?

You restrict the possible values of lambda_0 to (0,1), this way, the cut will be between m and C. and you're right about the method

awesome. Many thanks for your great help!