Suppose otherwise. The let a and b be points in disjoint components of f(A).
Let C be a curve connecting a to b in A, then f(C) is in f(A), but now f(C)
is a curve in R^m conecting f(a) to f(b), but this contradicts the assumption
that a and b are in disconnected components of f(A).
(here the notion of curve implies continuity, and the continuity of f ensures
that f(C) is a curve in the appropriate sense)