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Magazine Chapter 7: Problem 1
Let \(f\) be a bounded function on \([a ; b] .\) Then \(f\) is Riemann integrable if and only if \(\forall \varepsilon>0, \exists \delta>0\) such that for all partitions \(\mathscr{P}\) of \([\boldsymbol{a} ; \boldsymbol{b}]\), $$ \|\mathscr{P}\| \mid<\delta \Rightarrow U(f, \mathscr{P})-L(f, \mathscr{P})<\varepsilon $$
Short Answer
Expert verified
A function is Riemann integrable if upper and lower sums converge for fine partitions.
Step by step solution
01
Understanding the Exercise
The exercise we have here is about the definition of Riemann integrability of a bounded function on a closed interval. Specifically, the task is to prove that a function \(f\) is Riemann integrable if and only if for every positive \(\varepsilon\), there exists a \(\delta>0\) such that for all partitions \(\mathscr{P}\) of the interval \([a, b]\), having a norm smaller than \(\delta\) implies that the difference between the upper sum \(U(f, \mathscr{P})\) and the lower sum \(L(f, \mathscr{P})\) is less than \(\varepsilon\). This exercise involves understanding the partition norm and the difference between the Riemann upper and lower sums.
02
Definitions and Initial Conditions
First, let's clarify the definitions. A partition \(\mathscr{P}\) of \([a, b]\) is a set of points \(\{x_0, x_1, ..., x_n\}\) where \(a = x_0 < x_1 < ... < x_n = b\). The norm of the partition, given by \(\|\mathscr{P}\|\), is the length of the largest subinterval, i.e., \(\|\mathscr{P}\| = \max(x_{i} - x_{i-1})\). The upper sum \(U(f, \mathscr{P})\) is the sum over each subinterval of the product of its length and the supremum of \(f\) in that subinterval, and similarly the lower sum \(L(f, \mathscr{P})\) uses the infimum.
03
Forward Implication (Necessity)
Let us consider the forward implication, which is if \(f\) is Riemann integrable, for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that for all partitions \(\mathscr{P}\) of \([a, b]\) with \(\|\mathscr{P}\| < \delta\), we have \(U(f, \mathscr{P}) - L(f, \mathscr{P}) < \varepsilon\). Since \(f\) is Riemann integrable, by definition, the upper and lower Riemann sums converge to the same limit. Therefore, as the partition gets finer, \(U(f, \mathscr{P}) - L(f, \mathscr{P})\) becomes arbitrarily small. This provides the given \(\varepsilon-\delta\) condition.
04
Reverse Implication (Sufficiency)
Now, for the reverse implication, assume for every \(\varepsilon > 0\) there exists a \(\delta > 0\) such that for all partitions \(\mathscr{P}\) with \(\|\mathscr{P}\| < \delta\), \(U(f, \mathscr{P}) - L(f, \mathscr{P}) < \varepsilon\). This means that as partitions refine (i.e., norm reduces), the upper and lower sums converge closely. The condition implies that \(f\) can be uniformly approximated by step functions, and thus, converges to a limit. Hence, the function \(f\) is Riemann integrable on \([a, b]\).
05
Conclusion
By establishing both the forward and reverse implications, we have shown that \(f\) is Riemann integrable if and only if for every \(\varepsilon > 0\), there is some \(\delta > 0\) such that for any partition \(\mathscr{P}\) of \([a, b]\) with \(\|\mathscr{P}\| < \delta\), \(U(f, \mathscr{P}) - L(f, \mathscr{P}) < \varepsilon\). This concludes the proof of the criterion for Riemann integrability provided in the exercise.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Upper and Lower Sums
In Riemann integration, the concepts of upper and lower sums are pivotal. They help in approximating the area under a curve of a bounded function on a closed interval
- The upper sum is created by taking the maximum value of the function, known as the supremum, over each subinterval of a partition. We then multiply this supremum with the length of the subinterval and sum these products over the entire interval.
- Conversely, the lower sum relies on the minimum value, or infimum, over each partition subinterval. Again, this infimum is multiplied by the subinterval length and summed.
Partition Norm
A partition subdivides an interval
- It consists of a finite sequence of points beginning and ending at the interval's bounds.
- The partition norm refers to the largest gap or subinterval length between consecutive points in the partition.
Epsilon-Delta Criterion
The epsilon-delta criterion is a formal approach to determine Riemann integrability. This mathematical statement essentially describes how close one must get in approximating the integral of a function as a key condition for Riemann integrability.
- For any given small positive number, denoted as \( \varepsilon \), there exists a dependent 'threshold' \( \delta \).
- This \( \delta \) guarantees that when a partition's norm is less than \( \delta \), the difference between the upper and lower sums does not exceed \( \varepsilon \).
