# Isometric mapping of two subsets in a metric space

Let (M,d) be a metric space, and M1, M2 be two subspace of M.

Edit: I forgot to mension that both M1 and M2 are dense.

Suppose we have

$$f: M rightarrow M$$
and we know f is an isometric mapping from M1 to M1, i.e.,
$$d(f(a),f(b))=d(a,b)$$
for any $$a,b in M1$$.

Suppose we also have
$$d(f(a),f(b))=d(a,b)$$
for any $$a in M1$$ and $$b in M2$$.

The question is : can we say that f is an isometric mapping of $$M1cup M2$$?

In other words, is it possible to show that
$$d(f(a),f(b))=d(a,b)$$
for any $$a,b in M1cup M2$$.

I think it is true in Euclidean space, but I have no idea how to do it in a general metric space.

Any suggestion would be greatly appreciated.

Mathematics Asked by user856180 on December 28, 2020

$$M_1={a}$$, $$M_2={b,c}$$, $$f(a)=a, f(b)=f(c)=b$$, $$d(x,y)=1$$ if $$xneq y.$$

Answered by Tsemo Aristide on December 28, 2020

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