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Prove that a strictly monotonic function with range open set is continuous. I have proved for case when range is connected set.

Further question: How to prove f continuous when range is closed set.

Problem 4.65 Mathematical Analysis Apostol

Sushil
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1 Answers1

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For a monotonic function, how many possible discontinuities does it has?By the way, where is the CAP... box? The system ask me whether i'm a human being.

laboon
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  • there are countable no. of discontinuity. But how it helps. And discontinuity are only jump discontinuity. What is CAP...Box – Sushil Mar 19 '15 at 10:11
  • for an $x_0$ that is a discontinuous point, where should $f(x_0)$ be placed at the interval $[f(x_0-0), f(x_0+0)]$ to make the range still open? – laboon Mar 19 '15 at 10:19
  • Ok got the hint. One more question how to prove f continuous when range is closed – Sushil Mar 22 '15 at 08:27
  • You still may use the previous discussion. – laboon Mar 24 '15 at 11:53
  • Thanks a lot. took time but able to prove using only hint given by you – Sushil Mar 25 '15 at 17:38