You need to be aware that if f and g are both discontinuous at a point, it is possible for f+ g, f- g, f*g, and even f/g to be continuous at that point. Can you see why?
What is the possibility of continuity of combined function of
f+g, f-g, f*g, f/g at x=c, g(x) not equal to zero at x=c.
For all the combination possible
When f(x) Continuous & Discontinuous
g(x) Continuous & Discontinuous