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.