A function without a maximum and a minimum

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SUMMARY

The discussion centers on identifying a non-continuous function defined on the interval [0,1] that lacks both a maximum and a minimum. The proposed function is f(x) = 1/2 if x is a rational number and f(x) = x if x is an irrational number. This function is non-continuous due to the differing definitions for rational and irrational inputs, and it does not attain a maximum value as it approaches 1. The reasoning aligns with the principle that continuous functions on compact sets possess both maximum and minimum values.

PREREQUISITES
  • Understanding of non-continuous functions
  • Familiarity with rational and irrational numbers
  • Knowledge of compact sets in real analysis
  • Basic principles of limits and function behavior
NEXT STEPS
  • Study the properties of non-continuous functions in real analysis
  • Explore the concept of compactness in mathematical sets
  • Learn about the implications of the Extreme Value Theorem
  • Investigate other examples of functions that lack maximum and minimum values
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Students of mathematics, particularly those studying real analysis, and educators seeking examples of non-continuous functions for instructional purposes.

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Homework Statement



Give an example of a non-continuous function on [0,1] that has no maximum and no minimum.

Homework Equations



Well, a continuous function on a non-empty compact set will have a maximum and a minimum, so I guess this is why we need a non-continuous function.

The Attempt at a Solution



Does this work?

f(x) =

1/2 if x is a rational number
x if x is an irrational number.

Does this work? Breaking this up into 1/2s (if rational) and x's (if irrational) hopefully makes it non-continuous, and I suppose there is always some larger number that could be formed as we approach 1 (so no maximum). What do you think?
 
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I would agree.
 

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