Piecewise Continuity

A function is piecewise continuous on an interval if it consists of a finite number of continuous "pieces" that are joined together by simple, finite breaks.

It is the mathematical way of describing a function that is mostly smooth and well-behaved, but has a few sharp, sudden jumps.

1. The Two Strict Criteria

For a function $f(t)$ to be considered piecewise continuous on a closed interval $[a, b]$, it must meet two conditions:

Rule 1: Finite Number of Breaks

The interval can be divided into a finite number of subintervals (e.g., $[a, t_1], [t_1, t_2], \dots, [t_n, b]$) such that the function is completely continuous inside each subinterval. In plain terms: you can draw each individual piece without lifting your pen from the paper.

Rule 2: Only "Finite Jump" Discontinuities

At the boundaries where the pieces meet (the points of discontinuity), the function cannot shoot off to infinity.

If you look at a boundary point $t_c$, the limit as you approach from the left ($\lim_{t \to t_c^-} f(t)$) and the limit as you approach from the right ($\lim_{t \to t_c^+} f(t)$) must both exist as finite numbers.