Continuity 2.4

This function has discontinuities at x=1 and x=2. Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. 1 2 3 4 This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

Show g(x)=x^2 + 1 is continuous at x = 1

Types of Discontinuities There are 4 types of discontinuities Jump Point Essential Removable The first three are considered non removable

Jump Discontinuity Occurs when the curve breaks at a particular point and starts somewhere else Right hand limit does not equal left hand limit

Point Discontinuity Occurs when the curve has a “hole” because the function has a value that is off the curve at that point. Limit of f as x approaches x does not equal f(x)

Essential Discontinuity Occurs when curve has a vertical asymptote Limit dne due to asymptote

Removable Discontinuity Occurs when you have a rational expression with common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is removable.

Places to test for continuity Rational Expression Values that make denominator = 0 Piecewise Functions Changes in interval Absolute Value Functions Use piecewise definition and test changes in interval Step Functions Test jumps from 1 step to next.

Continuous Functions in their domains Polynomials Rational f(x)/g(x) if g(x) ≠0 Radical trig functions

Find and identify and points of discontinuity Non removable – jump discontinuity

Find and identify and points of discontinuity Non removable – essential discontinuity VA at x = 4

Find and identify and points of discontinuity 2 points of disc. (where denominator = 0) Removable disc. At x = 5 Non removable essential at x = -1 (VA at x = -1)

Find and identify and points of discontinuity Non removable point discontinuity

Find and identify and points of discontinuity 2 points of disc. (where denominator = 0) Removable disc. At x = 5 Non removable essential at x = -4 (VA at x = -4)