Can addition or subtraction of one continuous and other discontinuous function ever be continuous?

Answer 1

No.

To see why, suppose that this is possible. Let c(x) be continuous, d(x) be discontinuous, and suppose

f(x) = (c + d)(x) is also continuous. Then we must have

f(x) = c(x) + d(x) <----> f(x) - c(x) = d(x)

But here is where the problem lies. Because both f(x) and c(x) are continuous, the function (f - c)(x) is also continuous. This is a contradiction. We know d(x) is discontinuous and that d(x) = (f - c)(x).

Because we have a contradiction, our assumption that ((c + d)(x) is also continuous must be false.