f/g is continuous

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Here I show that the ratio of two continuous functions is continuous. I do it both by using epsilon-delta and the sequence definition of continuity. Interestingly, the proof is similar to the proof of the quotient rule for derivatives. Enjoy!

Dr Peyam

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Ooh nice. I've only seen the "Proof by inscrutable choice of delta" version of this. Very nice how you motivated all the steps.

martinepstein

I showed 1/x is continuous (the boring sequence way) then used the fact that the composition of continuous functions is continuous

thedoublehelix

I think you can show more easily the inequality than having to use the reverse triangle inequality.
If |f(x)-f(x0)|<|f(x0)|/2 then
-f(x0)|/2 <f(x)-f(x0)<f(x0)|/2 which means
f(x0)|/2<f(x)<(3/2)f(x0)|

Happy_Abe

f/g is continuous

fg is continuous

f/g is continuous

f+g is continuous

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