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# Continuous Functions watch

1. Hi,

I am stuck with this past paper question:

give the definition of a function f being continuous at x0, which I have done

then it says for the following three statements say if it is true or false and provide a counter example or proof:

1. if f and g are continuous at x0 then so is f + g

2. if f + g is continuous at x0 then so are f and g

3. if f and f + g are continuous at x0 then so is g

I can do 2. as if you let f(x) = 1 / x and g(x) = 1 - f(x), then f is discontinuous and f + g is continuous but im not sure about the other 2?

any one help?

thanks
2. 1 is true - you may be able to do this with an epsilon/2 argument but I'm not certain. It is definitely true though.
3. For #1 you can use the triangle inequality. That is, (think how to apply this here). For #3 you can do a similar thing, noting that .
4. (Original post by Big_Sam)
I can do 2. as if you let f(x) = 1 / x and g(x) = 1 - f(x), then f is discontinuous and f + g is continuous but im not sure about the other 2?
This isn't actually valid: f and g aren't defined at x = 0, (and therefore neither is f+g).
5. (Original post by DFranklin)
This isn't actually valid: f and g aren't defined at x = 0, (and therefore neither is f+g).
easily remedied by saying that f and g are both =1/2 at x=0.

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