Interactive Real Analysis

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 7.1. Riemann Integral
• 7.2. Integration Techniques
• 7.3. Measures
• 7.4. Lebesgue Integral
• 7.5. Riemann versus Lebesgue
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

7.3. Measures

Example 7.3.1(f): Oddities of Riemann Integral

There is a sequence of Riemann integrable functions f_n that converges to a function f that is not Riemann integrable.

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We will talk about sequences of functions, convergence, and the like in the next chapter, but it seems desirable that if a sequnce of functions converges and each member of the sequence has a certain property, then the limit function should also have that property.

Actually, this turns out to not be the case for many properties: you can find a sequence of continous functions whose limit exists but is not continuous, or a sequence of differentiable functions whose limit exists but is not differentiable, and similarly with integrability. The fault is actually with the definition of convergence, which we will introduce in the next chapter, but still, it would be nice if the limit of integrable functions was integrable.

If you want to jump ahead to see an example of this sequence, please click here.