Calculus- Early Transcendentals, 2021a

3.7. Continuity 105
Definition 3.43: Continuity on an Open Interval
A function f is continuous on an open interval (a,b) if it is continuous at every point in the
interval.
Furthermore, a function is everywhere continuous if it is continuous on the entire real number line
(−∞,∞).
Recall the function graphed in a previous section as shown in Figure 3.4.
Figure 3.4: A function with discontinuities at x = −5, x = −2, x = −1 and x = 4.
We can draw this function without lifting our pencil except at the points x = −5, x = −2, x = −1, and
x = 4. Thus, f (x) is continuous at every real number except at these four numbers. At x = −5, x = −2,
x = −1, and x = 4, the function f (x) is discontinuous.
At x = −2 wehavearemovable discontinuity because we could remove this discontinuity simply by
redefining f (−2) to be 3.5. At x = −5 andx = −1 wehavejump discontinuities because the function
jumps from one value to another. From the right of x = 4, we have an infinite discontinuity because the
function goes off to infinity.
Formally, we say f (x) has a removable discontinuity at x = a if lim x→a f (x) exists but is not equal to
f (a). Note that we do not require f (a) to be defined in this case, that is, a need not belong to the domain
of f (x).