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Continuity and Limits

Terms

Limits and Continuity

Defining a Limit

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.

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