Continuity Created by Mrs. King OCS Calculus Curriculum

Continuity at a Point on an Open Interval A continuous function is one which is unbroken along a given interval.

Continuity at a Point on an Open Interval Continuity may be destroyed at x=c if: 1.The function is not defined at x=c.

Continuity at a Point on an Open Interval Continuity may be destroyed at x=c if: 1.The function is not defined at x=c. 2.The limit of f(x) does not exist at x=c.

One-Sided Limits Only examining one side of the function. Limit from the left: Limit from the right:

One-Sided Limits For example: Limit from the left: Limit from the right:

Existence of a Limit 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 and

Continuity at a Point on an Open Interval Continuity may be destroyed at x=c if: 1.The function is not defined at x=c. 2.The limit of f(x) does not exist at x=c. 3.The limit of f(x) exists at x=c, but is not equal to f(c)

Continuity of a Function Examine the continuity of each function.

Properties of Continuity Created by Mrs. King OCS Calculus Curriculum

Properties If b is a real number and f and g are continuous at x=c, then the following functions are also continuous at c. Scalar multiple: bf Sum or difference: f ± g Product: fg Quotient: f/g, if g(c)  0

Composite Functions If g is continuous at c and f is continuous at g(c), then the composite function given by f(g(x)) is continuous at c.

One Last Point Types of functions which are continuous at every point in their domains. ▫Polynomial ▫Rational ▫Radical ▫Trigonometric