Barrons AP Calculus

have a discontinuity there. We cannot, however, “remove” a jump discontinuity
by any redefinition whatsoever.
Whenever the graph of a function f (x) has the line x = a as a vertical
asymptote, then f (x) becomes positively or negatively infinite as x → a+ or as x
→ a−. The function is then said to have an infinite discontinuity. See, for
example, Figure N2–4 for , Figure N2–5 for , or Figure N2–7

for . Each of these functions exhibits an infinite discontinuity.

Example 24 __

    is  not continuous  at  x   =   0   or  =   −1, since   the function    is  not

defined for either of these numbers. Note also that neither

exists.

Example 25 __

Discuss the continuity of f, as graphed in Figure N2–9.

SOLUTION: f (x) is continuous on [(0,1), (1,3), and (3,5)]. The discontinuity at
x = 1 is removable; the one at x = 3 is not. (Note that f is continuous from the
right at x = 0 and from the left at x = 5.)

Figure  N2–9

In Examples 26 through 31, we determine whether the functions are
continuous at the points specified: