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

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We prove that f(x)=x, the identity function, is continuous on its entire domain D, for any nonempty subset D of 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 of course is much more comprehensible in the video, where we can use mathematical notation. Let me know if there are more epsilon delta continuity proofs you want to see!

#Proofs #RealAnalysis

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Thanks for watching! Several more continuity proofs are coming soon!

WrathofMath

"You'll be happy to know that they don't get any easier from here." Haha. Short and sweet. Appreciated the time you spent at the end pointing out the interesting implication of the proof for f(x) = x .

AManWithaZ

can you please prove f(x)=5x is continues for all real numbers using epsilon-delta

HMH

can you prove 2^x is continuous using epsilon delta definition please?

楚阑

Sir please make a video on sequential continuity.

kavitaarya

How can we infur x is c from x is not c?

princez

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