Is an increasing monotonic function in a closed interval also continuous?

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SUMMARY

Every increasing monotonic function defined on a closed interval is not necessarily continuous. A counterexample is the piecewise function f(x) = x for 0 ≤ x ≤ 1 and f(x) = x + 1 for 1 ≤ x ≤ 2, which demonstrates that while the function is monotone increasing, it is discontinuous at every integer. This discussion highlights the importance of understanding the distinction between monotonicity and continuity in mathematical functions.

PREREQUISITES
  • Understanding of monotonic functions
  • Knowledge of continuity in mathematical analysis
  • Familiarity with piecewise functions
  • Basic concepts of closed intervals in real analysis
NEXT STEPS
  • Study the properties of monotonic functions in detail
  • Research the definitions and examples of discontinuous functions
  • Explore the implications of the Intermediate Value Theorem
  • Learn about the relationship between continuity and differentiability
USEFUL FOR

Mathematics students, educators, and anyone interested in real analysis or the properties of functions will benefit from this discussion.

Lancelot1
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Hello all,

Is this statement true ? Is every increasing monotonic function in a closed interval also continuous ?

How do you prove such a thing ?

Thank you !
 
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You don't prove it, it isn't true!

For example the function f(x)= x for [math]0\le x\le 1[/math], f(x)= x+1 for [math]1\le x\le 2[/math], and, in general, f(x)= x+ n for [math]n\le x\le n+1[/math] for n an integer is monotone increasing for all x but is discontinuous at every integer.
 

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