Stage 37 · Beyond (Optional)

37.5.5 Global Properties of Continuous Functions

37.5 Into Real Analysis: Making Limits and Continuity Rigorous

Point 5 of 5

37.5.5 Global Properties of Continuous Functions

Core idea

A function continuous on a closed interval must attain a maximum and minimum and must hit every value in between, with the intermediate value and extreme value theorems bringing the journey to a close.

Module goal. Return to the limits you met earlier and use the ε–δ language to say exactly what "getting infinitely close" means, closing out the whole path and opening the door to higher mathematics.

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