Reaalväärtused

Reaalväärtused refer to values that belong to the set of real numbers, denoted by R. In mathematics, an object is called real-valued if its outputs lie in R. The most common context is real-valued functions, where a function f maps elements from a domain X to real numbers, written as f: X → R, and for every input x, the value f(x) is a real number.

Examples include f(x) = x^2 for x ∈ R, where f(3) = 9, a real value; and f(x) = sin(x) for x ∈ R, where f(x) ∈ [−1, 1], always a real value. Real-valued objects also appear in other settings, such as real-valued sequences (a_n ∈ R for all n) and real-valued random variables in probability theory (X(ω) ∈ R for almost every outcome ω).

Real-valued is often contrasted with complex-valued or vector-valued. A complex-valued function, for example, has outputs in C, the set of complex numbers, while a real-valued function has outputs restricted to R. The term is common in Estonian mathematical discourse, where reaalarvud refers to real numbers and reaalväärtus to the value taken in R by a function, sequence, or random variable.

In practice, statements about real-valued functions emphasize properties and operations that remain within the real numbers, such as continuity, differentiability, integrability, and convergence, which are foundational in calculus and analysis.