◀ ▲ ▶Branches / Analysis / Definition: Riemann-Integrable Functions

The following definition makes use of the fact that for any real functions $h,f,g$ bounded on an interval $[a,b]$ it follows from $h(x)\le f(x)\le g(x)$ for all $x\in[a,b]$ that $\int_a^b h(x)dx\le \int_a^b f(x)dx\le\int_a^b g(x)dx.$ In particular, if $h,g$ are step functions then $$\int_a^{b} h(x)dx\le\underbrace{\int_{a~*}^{b} f(x)dx}_{=:L}\le \int_a^b f(x)dx\le\underbrace{\int_a^{b~*} f(x)dx}_{=:U}\le\int_{a}^{b} g(x)dx.$$ If the two lower and uppend bounds $L$ and $U$ are equal, then they must be also equal the integral in the middle.

Definition: Riemann-Integrable Functions

Let \([a,b]\) be a closed real interval. A bounded real function \(f:[a,b]\mapsto\mathbb R\) is called Riemann-integrable, if its Riemann upper and Riemann lower integrals are equal:

$$\int_{a}^{b~*}f(x)dx=\int_{a~*}^{b}f(x)dx.$$

In this case, we set

$$\int_{a}^{b}f(x)dx=\int_{a}^{b~*}f(x)dx.$$

Notes

• The expression $\int_{a}^{b}f(x)dx$ is called the Riemann integral of the function $f$ on the interval $[a,b].$
• We can use another variable instead of $x$ without changing the meaning and the value of the Rieman integral, e.g. $$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt=\int_{a}^{b}f(y)dy=\ldots$$

Table of Contents

Examples: 1

1. Proposition: Step Function on Closed Intervals are Riemann-Integrable
2. Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable
3. Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable
4. Proposition: Linearity and Monotony of the Riemann Integral
5. Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable
6. Proposition: Product of Riemann-integrable Functions is Riemann-integrable
7. Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions

Mentioned in:

Definitions: 1
Examples: 2
Lemmas: 3 4
Parts: 5
Proofs: 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Propositions: 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Subsections: 41 42
Theorems: 43 44 45 46

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983