Proof f(x)=sin(x) is Continuous using Epsilon Delta Definition | Real Analysis Exercises

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We prove that f(x)=sin x, the sine function, is continuous on its entire domain - the real numbers. We complete this proof using the epsilon delta definition of continuity of a function at a point. To do this, we simply take an epsilon greater than 0 and an arbitrary point c from our domain, then go through the motions of finding a delta greater than 0 so that any x in D that is within delta of c has an image within epsilon of c's image. This will be surprisingly easy so long as we remember a few important trigonometric identities/inequalities! Let me know if there are more epsilon delta continuity proofs you want to see!

#RealAnalysis #Math

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Also can be proved by limit and diffrentiation concepts. If the function is diffrentiable, then it is continous by defination. So, we can use [f(x+h)-f(x)]/h formula to prove that easily. I dont know these delta function and all as I am grade 12 student, but if I was asked to prove that, I would use this approach. Sry for my english

maheshpatel

Another way to do this, but not as good, would be to take the arcsin to get the "x - c" term to pop out. I say it's not as good because it involves a little more work, and even when you're done (it turns out that delta = 2*arcsin(epsilon/2) ), it's fair to then start wondering whether arcsin has any weird behavior that makes it a bad solution. I mean, it looks like it should satisfy the requirements of the original problem since arcsin(0) = 0, but a person can feel a lot more comfortable that a simple "delta = epsilon" will scale things down to nothing as desired. You also have to account for periodicity; hardly insurmountable, but it's still a needless hassle.

I am sort of working on a mental Foolproof Guide To Epsilon-Delta; obviously nothing is foolproof here. But I feel like the first milestone to success is to find a way to pop out a term of |x - c|. Once you've done that, find a way to get rid of any remaining x's, probably by applying "this must be less than that" logic and possibly constraining delta as part of that logic. Once you've done that, unceremoniously shove everything other than the |x - c| to the other side, and then you're done. Maybe?

kingbeauregard

your sound like sheldon cooper of big bang theory.... plus great explanation ❤

codingnewbie

Great video. I might be nitpicky but |sin(x)|<=|x| seems very small angle approximation'ish to me but it's been awhile since i've seen the simple derivation for arc length = radius * subtended angle. Nice using triangles though. I usually like to think for the cosine one just in terms of bounds of the periodic graph of sine.

theproofessayist

Hey can you show us a really hard epsilon-delta? Like suppose the function is 1 / [(x+1)(x+2)(x+3)]; that's got to be super duper difficult to epsilon-delta into submission, right? Unless ... it's actually tons easier than it looks?

kingbeauregard

Since it is also true that |sin(x)|<=1, can you just set delta = 2?

adoxographer

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