Let F: D->R and suppose that f(x)>= 0 for all x elemend of D. Define sqrt(f): D->R by sqrt(f)(x) = sqrt(f(x)). If f is continuous at c element of D, prove that sqrt(f) is continuous at c.
Proof 1] If f is continous at x_0 and g is continous at g(x_0) then g o f is continous at x_0.
Sine f(x) is continous and g(x)=sqrt(x) is continous so if their composition.
Proof 2]Let {x_n} be a sequence in D converging to x_0. Then lim f(x_n) = f(x_0) by definition of continuity.
But then,
lim sqrt(f(x_n)) = sqrt(x_0) = sqrt(f(x_0))
Q.E.D.