# Section 1.4.  “open interval”- an interval where the function values at the endpoints are undefined (a, b)  “closed interval”- an interval where the.

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Section 1.4

 “open interval”- an interval where the function values at the endpoints are undefined (a, b)  “closed interval”- an interval where the function values at the endpoints are defined [a, b]  can be mixed: (a, b] [a, b)

 a function (or part of one) is considered to be “continuous” when it has no “holes” (undefined values), jumps, or gaps (no interruptions)  when a function (or part of one) is considered to be “discontinuous”, there are two types: --- “removable”: when the discontinuous part of the function can be made to be continuous simply by redefining the function in some way --- “nonremovable”: when nothing can be done to the function to make the discontinuous part continuous

 Continuity of a function at a point: --- for a function to be continuous at a given point, the following conditions must be satisfied: 1) the function value is defined at the point 2) the limit as x approaches the point must exist 3) the function value and the limit must be equal

 Continuity on an open interval: --- for a function to be considered continuous over an open interval, it must be continuous at every point within the interval (if continuous over the entire number line, it is considered to be continuous “everywhere”)

 “two-sided” limits are limits that are determined when both the left and right sides of a function approach a particular value  “one-sided” limits are limits that are determined when only one side of a function approaches a particular value --- limit from the “left” and limit from the “right” --- denoted by either a little + or – next to the approaching x-value depending on the side (--: left side, +: right side)

 If both one-sided limits are equal, then a two- sided limit exists and is that limit…

 Continuity on a closed interval: --- for a function to be considered continuous on a closed interval, the limit as x approaches from the left must equal the function value of the right endpoint, and the limit as x approaches from the right must equal the function value of the left endpoint

 If the functions f(x) and g(x) are continuous, then the following properties exist: 1) scalar multiple: b(f(x)) is continuous 2) sum and difference: f(x) ± g(x) is continuous 3) product: f(x)g(x) is continuous 4) quotient: f(x)/g(x) is continuous (if g(x) ≠ 0)

 If f(x) and g(x) are continuous, then both f(g(x)) and g(f(x)) are continuous as well…

 applies to functions that are continuous on closed intervals  says that if a number (function value) is between the function values of the two endpoints, then there exists at least one value of x that when substituted into the function, that number (function value) will be the result  also can be called an “existence theorem”

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