I know that the following are false, I just cant come up with counterexamples:

[f(x)]^2 continuous on (a,b) implies taht f is continuous on (a,b)

f and g are not continuous on (a,b) implies that fg is not continuous on (a,b)

f and g are not continuous on (a,b) implies that $\displaystyle f\circ g$ is not continuous on (a,b)