As seen already in Fig. 8.2, it can happen that both the left-hand limit and the right-hand limit of a function at x=a exist as finite numbers, but the two numbers are different from each other — so the graph visibly "jumps" as x crosses a. This situation is called a jump discontinuity:
A function f(x) has a Jump Discontinuity at x=a if limx→a−f(x) and limx→a+f(x) both exist but limx→a−f(x)=limx→a+f(x).
Illustration 5. Consider f(x)=x2−x−5 for −4≤x<−2, and f(x)=x3−4x−3 for −2≤x≤1. At x=−2, f(−2)=(−2)3−4(−2)−3=−8+8−3=−3 (using the second formula, since −2 belongs to that branch). Now
limx→−2−f(x)=limx→−2(x2−x−5)=4+2−5=1,limx→−2+f(x)=limx→−2(x3−4x−3)=−8+8−3=−3. …