continuity

The concept of a _limit_ is fundamental in mathematics. Intuitively, the limit of a function \( f(x) \) as \( x \) approaches a point \( x_0 \) allows us to analyze the behavior of the function as the values of \( x \) get arbitrarily close to \( x_0 \).

The Intermediate Value Theorem states that if a function is continuous on a closed interval, then it takes every value between its values at the endpoints.

A function is continuous if the limit at each point exists and coincides with the function’s value, ensuring no abrupt changes in its behavior.

Ecco una proposta per l’excerpt, descrittiva e in stile più manualistico, senza “this guide”:

Discontinuity of a real function is the failure of continuity at a point, occurring when the limit does not exist, is not finite, or does not coincide with the function value. The main types include removable, jump, and infinite discontinuities, each reflecting a distinct way in which local behavior may break down.

Uniform continuity is a stronger form of continuity where a single \(\delta\) works for the entire domain, ensuring that small changes in \(x\) produce uniformly small changes in \(f(x)\).

Piecewise functions are functions defined by different formulas on different parts of their domain, with each formula applying under specified conditions.

The absolute value function assigns to each real number its distance from zero on the number line:

\[
y = |x| =
\begin{cases}
+x & \text{if } x \geq 0 \\[0.5em]
-x & \text{if } x < 0 \end{cases} \]