18.100B Real Analysis
18.100B Real Analysis (Spring 2025, MIT OCW). Instructor: Prof. Tobias Holck Colding. This course gives an introduction to analysis, and the goal is twofold:
1. To learn how to prove mathematical theorems in analysis and how to write proofs.
2. To prove theorems in calculus in a rigorous way.
The course will start with real numbers, limits, convergence, series and continuity. We will continue on with metric spaces, differentiation and Riemann integrals. After that, we will move on to differential equations.
(from ocw.mit.edu)
| Lecture 18 - Integrable Functions |
We will show that continuous functions are integrable. This means that the graph of such a function bounds a well-defined area. To do so, we define what it means for a function to be uniformly continuous. This is a strong version of continuity but we will see that all continuous functions on a closed and bounded interval have this stronger property. Once we have shown that all continuous functions on a compact interval are uniformly continuous, it will follow relatively easily that they are integrable.
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