Terms
Continuity
-
A function is continuous at a point if the three following conditions are met:
1)
f (a) is defined. 2)
f (x) exists. 3)
f (x) = f (a). A conceptual way to describe continuity is this: A function is
continuous if its graph can be traced with a pen without lifting the pen from
the page.
Infinite Discontinuity
-
A category of discontinuity in which a vertical
asymptote exists at x = a and f (a) is
undefined.
Jump Discontinuity
-
A category of discontinuity in which
f (x)≠f (x), but both of these limits exist and are finite.
Limit
-
The value A to which a function f (x) gets arbitrarily close as the value of
the independent variable x gets arbitrarily close to a given value a. Such
a limit is symbolized this way: limx→af (x) = A.
One-Sided Limit
-
A limit based entirely on the values of a function taken at an
x-value
slightly greater than or less than a given value. Whereas a two-sided limit
f (x) takes into account the values of
x near
a which are
both greater than and less than
a, a one-sided limit from the left
f (x) or from the right
f (x) takes into account only
values of
x less than
a, or greater than
a, respectively.
Point Discontinuity
-
A category of discontinuity in which a function has a well-defined two-sided
limit at x = a, but either f (x) is not defined at a or its value at
a is not equal to this limit.
f (x) exists. 3) 
f (x)≠
f (x), but both of these limits exist and are finite.
