Can Two Discontinuous Functions Sum to a Continuous Function?

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Two discontinuous functions can indeed sum to a continuous function, as illustrated by the example of f(x) and g(x). Here, f(x) is defined as 0 for all x except 0, where it equals 1, while g(x) is 0 for all x except 0, where it equals -1. The sum of these functions, f(x) + g(x), results in a function that is continuous everywhere. This demonstrates that the sum can eliminate the discontinuities present in the individual functions. Understanding these concepts is crucial in exploring the behavior of discontinuous functions in mathematics.
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can you give me an example of two discontinuous functions at a number a whose sum is not discontinuous at a? :confused: thanks!:shy:
 
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i was just about to ask a question about dis. functions, and express each function as a composition, and i don't know how to do that??

if you give me a prob, i might help to see what you mean.
 
huh?

If the functions are discontinuous at a, then their sums are not going to be continuous at a. I suppose you could have two step functions, one going up and one going down, but I don't think that is really fair because there is still a discontinunity at a... Mathematicians?
 
Let f(x)=0 when x is not zero, and 1 when x is zero.
Let g(x)=0 when x is not zero, and -1 when x is zero.

then f+g is continuous everywhere.
 

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