Thursday, 30 April 2015

real analysis - Let f:DrightarrowmathbbR and assume that x0inD is not an accumulation point of D. Prove that f is continuous at x0.

(professor hints)



A Road Map to Glory




  • Write Down the negation of the definition of an accumulation point.

  • Prove that there exists a positive real number δ for which (x0δ,x0+δ)D={x0}

  • Prove that the only number x satisfying xD and |xx0|<δ is x=x0.

  • Prove that for such an x, |f(x)f(x0)|<ϵ for every positive number ϵ







I have trouble starting from the second bullet point. After that I wouldn't know how to connect it with the third. I wanted to ask for help regarding these two bullets. I understand that due to the negation of accumulation point there exists a finite neighborhood of x0.Im not sure how this connects to the third bullet. I understand that it the δ-neighborhood of x0, however how is that neighborhood finite.

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