A regulated function is a real‑valued function defined on a closed interval $[a,b]\subset\mathbb{R}$ (or, more generally, on any compact interval of the real line) that possesses a finite left‑hand limit and a finite right‑hand limit at every point of its domain. Formally, a function $f:[a,b]\to\mathbb{R}$ is regulated if for every $x\in(a,b)$ the limits
$$ \lim_{t\to x^-} f(t)\quad\text{and}\quad \lim_{t\to x^+} f(t) $$
exist (and are finite), and at the endpoints the one‑sided limits exist:
$$ \lim_{t\to a^+} f(t),\qquad \lim_{t\to b^-} f(t). $$
Main properties
| Property | Description |
|---|---|
| Uniform approximation by step functions | Every regulated function can be uniformly approximated by a sequence of step (simple) functions. This is a key characterization: $f$ is regulated ⇔ there exists a sequence $(s_n)$ of step functions with $|f-s_n|_\infty\to 0$. |
| Boundedness | On a compact interval, a regulated function is necessarily bounded, because the existence of one‑sided limits precludes unbounded oscillations. |
| Integrability | Regulated functions are Riemann‑integrable. The uniform approximation by step functions shows that the Riemann sums converge to a limit, furnishing the integral of $f$. |
| Closure under algebraic operations | The sum, product, and scalar multiples of regulated functions are regulated. The pointwise limit of a uniformly convergent sequence of regulated functions is regulated. |
| Relationship to continuity | Every continuous function is regulated, but the converse is false. Functions with jump discontinuities (e.g., the Heaviside step function) are regulated. |
Examples
- Continuous functions – any function $f$ continuous on $[a,b]$ trivially satisfies the definition.
- Step functions – functions that are constant on a finite collection of subintervals of $[a,b]$ and have a finite number of jumps.
- Monotone functions – every monotone function on $[a,b]$ has one‑sided limits everywhere and is therefore regulated.
- Functions with countably many jump discontinuities – for instance, $f(x)=\sum_{n=1}^{\infty} 2^{-n},\mathbf{1}_{{x\geq c_n}}$ where $(c_n)$ is a sequence of distinct points in $[a,b]$.
Historical context
The class of regulated functions was introduced in the early 20th century as part of the development of the Riemann integral and later as a convenient setting for the study of differential equations and functional analysis. The term “regulated” reflects the idea that the function’s behaviour is “controlled” by the existence of one‑sided limits at every point.
Key references that treat regulated functions include:
- J. Lebesgue, Leçons sur l’intégration et le problème des bases, 1904.
- L. C. Evans, Partial Differential Equations, 2nd ed., 2010 – Section on functions of bounded variation and regulated functions.
- R. G. Bartle, The Elements of Real Analysis, 2nd ed., 1976 – Chapter on Riemann integration.
Relation to other function classes
- Functions of bounded variation (BV): Every function of bounded variation on $[a,b]$ is regulated, but not every regulated function has bounded variation.
- Henstock–Kurzweil integrable functions: The Henstock–Kurzweil integral extends the Riemann integral and integrates all regulated functions; however, it also integrates many functions that are not regulated.
- Darboux functions: Functions whose derivative (in the sense of the Darboux property) need not be regulated; the classes are distinct.
Summary
Regulated functions form a broad class encompassing continuous functions, step functions, monotone functions, and many functions with jump discontinuities. Their defining feature—the existence of finite one‑sided limits at every point—ensures boundedness and Riemann integrability, while also allowing uniform approximation by simple step functions. This makes them a useful tool in classical analysis, integration theory, and the study of differential equations.
