The graph of f is shown above.
We observe that f is not continuous at x = −2, x = 0, or x = 2.
At x = −2, f is not defined.
At x = 0, f is defined; in fact, f (0) = 2. However, since and does not
exist. Where the left- and right-hand limits exist, but are different, the function has a jump
discontinuity. The greatest-integer (or step) function, y = [x], has a jump discontinuity at every
integer.
At x = 2, f is defined; in fact, f (2) = 0. Also, the limit exists. However,
This discontinuity is called removable. If we were to redefine the function at x = 2 to be f (2) = −2,
the new function would no longer 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 nor 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