let (X,d) be compact metric space and f:X --> X ISOMETRY PROVE f IS HOMORPHISM?

As f is isometry, then we have
d(f(x), f(y)) = d(x, y)
for x, y ∈ X.
f is continuous:
Now for given � > 0, if we select δ > 0 such that δ < �. Then we have:
Whenever d(x, y) < δ,
d(f(x), f(y)) = d(x, y) < δ < �
This proves that f is continuous.
f is injective:
Now suppose f(x) = f(y). Hence we have d(f(x), f(y)) = 0, this...