Elementary functions are continuous

Before looking for the points of discontinuity, it is worth knowing where they cannot be found: in the great majority of cases the functions we meet are continuous everywhere they are defined.

Remark — "Normal" functions are continuous

The elementary functions — polynomials, $e^x$, $\ln x$, $\sin x$, $\cos x$, $\tan x$, roots, powers — are all continuous where they are defined. The operations (sum, product, quotient with denominator $\ne 0$, composition) preserve continuity. Therefore every function obtained by combining elementary functions is continuous on its natural domain. Points of discontinuity can occur only:

• where the function changes formula (piecewise functions);
• at the edges of the domain (where the point is “missing”).

In practice, when studying a function, one only checks these two types of “suspect” points: the junction points of piecewise-defined functions and the points where the domain breaks off. Everywhere else continuity is guaranteed.

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Topics: Continuita
Concepts: Continuita · Funzione definita a tratti · Funzioni elementari