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

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The discussion centers on demonstrating that the max-min theorem fails for non-continuous functions. A specific example provided is the function f(x)=1/(x-a) for x not equal to a and f(x)=0 for x=a. This function illustrates that, despite being defined on the interval [a,b], it does not attain an absolute maximum or minimum due to its discontinuity at x=a. Participants are encouraged to visualize this with a graph to better understand the behavior of the function. The conclusion emphasizes that without continuity, the guarantees of the max-min theorem do not apply.
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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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