You are correct that if f1 is continous at, say, x= a, and f2 is not, then f cannot be continuous at x= a. But why are you raising the question? This statement says nothing about "f1 is continuous f2 is not". It simply says that if f is continuous then f1 and f2 can both be chosen to be continuous. Here it is fairly standard to take f1= (f(x)+ f(-x))/2 and f2= (f(x)- f(-x))/2. If f is continuous, so are f1 and f2. If f is differentiable, so are f1 and f2.