nellintervallo

Nell'intervallo is an Italian phrase used in mathematical writing to indicate that a statement applies to all values of a variable within a specified interval. The expression translates roughly as “in the interval” or “within the interval” and is a common linguistic convention in textbooks, lecture notes, and papers written in Italian. It is not a formal operator, but a way to convey domain restrictions without repeating the quantifier for every instance.

Usage and examples: For instance, one might encounter sentences such as “La funzione f è continua nell'intervallo [a,b]” or “La funzione è definita nell'intervallo (a,b).” In proofs, the phrase helps delimit where certain properties hold, so that the reader understands that the stated result is valid only for x within the specified bounds.

Relation to formal statements: To ensure precision, authors often accompany the phrase with explicit quantifiers, for example: “Per ogni x ∈ [a,b], P(x) è vera.” The combination of the natural-language expression nell'intervallo and the formal notation reinforces the intended scope of a result or condition.

Variants and scope: The term can be qualified further with inclusivity or exclusivity of endpoints, such as nell'intervallo chiuso [a,b] or nell'intervallo aperto (a,b). It may also appear in discussions of domains for functions, integrals, limits, and convergence, where the behavior is analyzed only for x within a specified range.

See also: Intervallo, continuità, derivabilità, integrazione.