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In secondary/high school students are rarely shown the proof to anything or told the limits of the methods they use. For example, integration of ‘proven’ by showing how the area under a curve can be broken up into rectangles and summed up. This is the foundation of Riemannian integration but the full proof is never given. Furthermore, it often gives the students the impression any function can be integrated, just like they might think any function can be differentiated. Undergraduate courses in analysis are to

The Cambridge undergraduate mathematics degree includes an analysis course in the first year. I do not have a pdf to hand which I or another student wrote when sitting the course, partly because the lecturer provided printed notes of a sort. Said notes can be obtained from the website of said lecturer, Professor Korner , they are called 1A Analysis. Here is a direct link to the pdf version of the notes. They cover the follow;

  • Series and sums
  • Continuity and differentiability
  • Various theorems associated to the above
  • Complex extensions of these theorems and principles
  • Standard or commonly considered functions and their properties
  • Riemannian integration

This is a 1st year lecture course.

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