**
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:
*lim*
_{xâÜ’a}
*f* (*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.