MAT.126
1.4-1.5

To say that a function f is continuous at x=c means that there is no interruption in the graph of f at c.That is, its graph is unbroken at c and there are no holes, jumps, or gaps.

A function f is continuous at c if the following three conditions are met.1. f(c) is defined.2. The limit of f(x) as x appraches c exists.3. The function value and the limit value are equal.

A function is continuous on an open interval (a,b) if it is continuous at each point in the interval.A function that is continuous on the entire real line is everywhere continuous.

Consider a function f, defined on an open interval I containing c (except possibly at c). This function is said to be discontinuous at c if any one of the following are true:The function f is not defined at x=c.The limit of f(x) does not exist at x=c.The limit of f(x) exists at x=c, but it is not equal to f(c).

A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining) f(c).

A discontinuity at c is called nonremovable if f cannot be made continuous by appropriately defining (or redefining) f(c).

Also called a right-hand limit.Refers to a limit where x approaches c only from values greater than c (to the right of c on a number line or on the x-axis).

Also called a left-hand limit.Refers to a limit where x approaches c only from values less than c (to the left of c on a number line or on the x-axis).

Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if the limit from the left of f(x) as x approaches c is L and the limit from the right of f(x) as x approaches c is L.

A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and the limit from the right of f(x) as x approaches a is equal to f(a) and the limit from the left of f(x) as x approaches b is equal to f(b).The function f is continuous from the right at a and continuous from the left at b.

See your text, p. 75

PolynomialRationalRadicalTrigonometric

See your text, p. 75

If f is continuous on the closed interval [a,b], f(a) is not equal to f(b), and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k.The method of bisection depends on the intermediate value theorem and therefore on the continuity of polynomials (and other functions).This is an example of an existence theorem, a theorem that states that something exists but does not provide a method for finding that which does exit.

Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself).The statement the limit of f(x) as x approaches c equals positive infinity means that for each M>0 there exists a delta>0 such that f(x)>M whenever 0<|x-c|<delta.The statement the limit of f(x) as x approaches c equals negative infinity means that for each N<0 there exists a delta>0 such that f(x)<N whenever 0<|x-c|<delta.To define the infinite limit from the left, replace 0<|x-c|<delta by c-delta<x<c.To define the infinite limit from the right, replace 0<|x-c|<delta by c<x<c+delta.

If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f.

Let f and g be continuous on an open interval containing c. If f(c) is not zero, g(c) is zero, and there exists an open interval containing c such that g(x) is not 0 for all x not c in the interval, then the graph of the function given byh(x) = f(x) / g(x)has a vertical asymptote at x=c.