CONTINUITY AND ONE-SIDED LIMITS Section 1.4

When you are done with your homework, you should be able to… Determine continuity at a point and continuity on an open interval Determine one-sided limits and continuity on a closed interval Use properties of continuity Understand and use the Intermediate Value Theorem

CONTINUITY AT A POINT A function is continuous at c if the following three conditions are met. is defined. exists.

CONTINUITY ON AN OPEN INTERVAL A function is continuous on an open interval if it is continuous at each point in the interval. A function that is continuous on the entire real line is everywhere continuous.

Is the following function continuous at x = 3? Yes No

Is the following function continuous at x = 2? Yes No since the limit as x approaches 2 does not exist. No since the function evaluated at 2 is not equal to the limit as x approaches 2.

ONE-SIDED LIMITS means the limit as x approaches c from the right. means the limit as x approaches c from the left. One-sided limits are useful in taking limits of functions involving radicals. If n is an even integer,

THE GREATEST INTEGER FUNCTION The greatest integer function is denoted

Evaluate -1 -2 Does not exist

THE EXISTENCE OF A LIMIT Let f be a function and let c and L be real numbers. The limit of as x approaches c is L if and only if and

Evaluate -1 0.5 Does not exist

CONTINUITY ON A CLOSED INTERVAL A function f is continuous on a closed interval if it is continuous on the open interval and and . The function f is continuous from the right at a and continuous from the left at b.

PROPERTIES OF CONTINUITY If b is a real number and f and g are continuous at , then the following functions are also continuous at c. Scalar multiple Sum and difference Product quotient

Polynomial Rational Radical Trigonometric EXAMPLES OF FUNCTIONS WHICH ARE CONTINUOUS AT EVERY POINT IN THEIR DOMAINS Polynomial Rational Radical Trigonometric

CONTINUITY OF A COMPOSITE FUNCTION If g is continuous at c and f is continuous at , then the composite function given by is continuous at c.

THE INTERMEDIATE VALUE THEOREM If f is continuous on the closed interval and d is any number between and , then there is at least one number c in such that

A VISUAL OF THE IVT

CONSIDER THE FUNCTION BELOW ON THE GIVEN INTERVAL CONSIDER THE FUNCTION BELOW ON THE GIVEN INTERVAL. IF THE IVT CAN BE APPLIED USE IT TO FIND THE INPUT VALUE WHICH YIELDS AN OUTPUT OF ½. The IVT cannot be applied since the function is not continuous. The IVT can be applied since the function is continuous on the given interval.