Summary of step functions and Riemann integration

Step function. Constant on the intervals of a subdivision of $[a,b]$.

Integral. $\int_a^bf=\sum_kc_k\Delta_k$ (independent of the subdivision).

Properties.
- Linearity: $\int(f+\lambda g)=\int f+\lambda\int g$.
- Additivity (Chasles): $\int_a^cf=\int_a^bf+\int_b^cf$.
- Monotonicity: $f\le g\Rightarrow\int f\le\int g$.
- Triangle inequality: $|\int f|\le\int|f|$.
- Zero integral: $f\ge0$ and $\int f=0\Rightarrow f$ vanishes at continuity points.

Darboux sums. For step functions, lower and upper sums coincide; step functions are Riemann integrable.

Squeeze criterion. $f$ is Riemann integrable iff for every $\varepsilon>0$ there exist step functions $\varphi\le f\le\psi$ with $\int(\psi-\varphi)<\varepsilon$.

Applications. Monotone functions and continuous functions are Riemann integrable (approximated by step functions).

Composition. $\Phi\circ f$ is Riemann integrable when $f$ is and $\Phi$ is continuous.

Conclusion. Step functions are the elementary building blocks of the Riemann integral; integrability is captured by their approximation.