If I have f(x) is continuous, and g(x) is not continuous, is it possible for f(x) + g(x) to be continuous?

Question

Nothing more to add :)

Answer 1

Hi Ashley,

 

Let's form function h(x) = f(x) + g(x), where f(x) is continuous and g(x) is not continuous on some interval [a, b]. A continuous function is one for which sufficiently small changes in x results in arbitrarily small changes in the output h(x). Given the discontinuous nature of g(x), it will generally affect h(x) in a manner that it will also generally be discontinuous.

 

You can also be more specific and use the concept of a limit. Let's say that function g(x) is not continuous at x = c. lim_{x -> c+}g(x) /=  lim_{x -> c-}g(x). This is my attempt of saying that the limit as x approaches c from -x is not the same as the limit as x approaches c from +x. This behavior will manifest itself for h(x) since the limit is a linear operation. I mean, lim_{x -> c}h(x) = lim_{x -> c+}f(x) +  lim_{x -> c+}g(x).

 

I hope this helps.

Answer 2

NO, BOTH FUNCTIONS MUST BE CONTINUOUS (OVER SOME STATED DOMAIN) FOR the sum to be continuous over that domain.