Barrons AP Calculus

(Marvins-Underground-K-12) #1

everywhere.
The trigonometric, inverse trigonometric, exponential, and logarithmic
functions are continuous at each point in their domains.
Functions of the type (where n is a positive integer ≥ 2) are continuous at
each x for which is defined.
The greatest-integer function f (x) = [x] (Figure N2–1) is discontinuous at each
integer, since it does not have a limit at any integer.


Kinds of Discontinuities

In Example 2, y = f (x) is defined as follows:


The graph   of  f   is  shown.

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

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