Graphical Proof: The Max-Min Theorem Does Not Hold for Non-Continuous Functions

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SUMMARY

The discussion centers on the Max-Min Theorem, specifically addressing its failure for non-continuous functions. The example provided is the function f(x) = 1/(x-a) for x ≠ a and f(x) = 0 for x = a, illustrating that this function does not attain an absolute maximum or minimum on the interval [a, b]. The graphical representation of this function clearly demonstrates the absence of continuity, which is crucial for the Max-Min Theorem to hold true.

PREREQUISITES
  • Understanding of the Max-Min Theorem in calculus
  • Knowledge of continuous vs. non-continuous functions
  • Familiarity with graphical representation of functions
  • Basic concepts of limits and discontinuities in real analysis
NEXT STEPS
  • Study the implications of discontinuities on function behavior
  • Learn about the properties of continuous functions and their importance in calculus
  • Explore graphical techniques for illustrating function behavior
  • Investigate other theorems related to extrema in calculus
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Students in calculus or real analysis, educators teaching the Max-Min Theorem, and anyone interested in the properties of continuous and non-continuous functions.

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Hi, just trying to do some homework for college and I can't get my head around this question. It is a question to show that if you take away the condition of a function being continuous, the max-min theorem no longer holds true. Any help is greatly appreciated!

Suppose that f: [a,b] -> R is not continuous. Show that f need not have an absolute maximum and an absolute minimum on [a,b]. (Answer in graphical form)
 
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Tell me about f(x)=1/(x-a) if x not equal to a and f(x)=0 if x=a. Then graph it.
 

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