Section 2.5 Continuity "Don't judge each day by the harvest you reap, but by the seeds you plant."        -Robert Louis Stevenson  

A function f is continuous at a number a if 3 implicit requirements: is defined (that is, a is in the domain of f) exists

At which numbers is f discontinuous? Why?

Example 2 Where are each of the following functions discontinuous?

Discontinuities: Removable Removable Infinite Discontinuity Jump Discontinuities

Definition: A function f is continuous from the right at a number a if And f is continuous from the left at a if

Example 3 At each integer n, the function is continuous from the right but discontinuous from the left because?

Definition: A function f is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)

Theorem: If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:

Theorem: Any polynomial is continuous everywhere; that is, it is continuous on Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

Example 5 Find

The following types of functions are continuous at every number in their domains: Polynomials Rational functions Root functions Trigonometric functions Inverse trigonometric functions Exponential functions Logarithmic functions

Example 6 Where is the function Continuous?

Example 7 Evaluate

Theorem If f is continuous at b and Then In other words, Basically, we get to use direct substitution for composite functions...

Example 8 Evaluate The Trick!

Theorem If g is continuous at a and f is continuous at Then the composite function Given by is continuous at a

Example 9 Where are the following functions discontinuous?

The Intermediate Value Theorem Suppose that f is continuous on the closed interval And let N be any number between f(a) and f(b), Where Then there exists a number c in such that

Example 10 Show that there is a root of the equation Between 1 and 2

Example 11 For what value of the constant c is the function f continuous on

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