# Proposition: Riemann Integral for Step Functions

Let $$\phi\in T[a,b]$$ be a step function with respect to the partition $$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$$, i.e. let $$\phi$$ take some constant values $$c_i$$ on the open intervals $$(x_{i-1},x_i)$$ $\phi (x)=c_i\quad\quad\text{for all } x\in(x_{i-1},x_i),\quad i=1,\ldots,n.$ The Riemann integral of the step function is defined as the sum. $\int_a^b\phi(x)dx:=\sum_{i=1}^nc_i(x_i-x_{i-1}).$

The Riemann integral of the step function is well-defined, i.e. its definition does not depend on the specific choice of the partition $$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$$.

The Riemann integral can be interpreted as the area between the $$x$$-axis and the step function. The area is counted positive, if it is above the $$x$$-axis (green) and negative if it is below the $$x$$-axis (red):

Proofs: 1

Proofs: 1 2
Propositions: 3 4

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### References

#### Bibliography

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