Section 2 2 Objective To understand the meaning

Section 2. 2 Objective: To understand the meaning of continuous functions. Determine whether a function is continuous or discontinuous. To identify the points where a given function is discontinuous.

Discontinuity 3. 1. 2. 3. 4. Types of discontinuity Hole in graph – removable discontinuity – the function is not defined at x=c Jump discontinuity Infinite discontinuity Oscillating discontinuity

Discontinuity 4. A function is continuous at x = c means that there is no interruption in the graph at c. The graph is unbroken at c; no holes, jumps , or gaps. 5. Polynomials are continuous at every real number. f(x) = anxn + an-1 xn-1 + ……. . a 2 x 2 + ax + c where n is a nonnegative integer

Class Examples Determine whether a function is continuous or discontinuous. Identify the pionts where a given function is discontinuous. 1.

Section 3. 5 Class Examples 1.

Section 3. 5 Class Examples 1. At x = 1 and x = 3 denominator equal zero At x = 1 and x = 3 vertical asymptotes At x = 1 and x = 3 infinite discontiuities , nonremovable

Class examples Determine whether a function is continuous or discontinuous. Identify the points where a given function is discontinuous. 2. X=2 infinite discontinuity nonremovable

Class example # 3 (a) Infinite discontinuity • Vertical Asymptote • Where denominator equals zero Discontinuous at x = -7 Discontinuous at x = - 3 Infinite discontinuity, nonremovable

4(a) 4 a X = -3 infinite discontinuity, nonremovable

Class examples # 4 (b) Infinite discontinuity §Vertical Asymptote §Where denominator equals zero

Class examples # 4(b). Infinite discontinuity §Vertical Asymptote §Where denominator equals zero X= 2, -2 infinite discontinuities, nonremovable

Class example # 4(a) Removable discontinuity • Hole • When you cancel a factor of the denominator with a factor of the numerator

Class example # 6 Removable discontinuity • Hole • When you cancel a factor of the denominator with a factor of the numerator X =2 hole discontinuity, removable

Class Examples 5(b). X=-3 hole, removable X=5 infinite discontinuity, nonremovable

Class example # 6 • Identify points of discontinuity X=-2, hole, removable X=2 infinite discontinuity, nonremovable

Jump Discontinuity 7(a) Different values from the left and the right sides Piecewise function

Jump Discontinuity 7(b) Different values from the left and the right sides Piecewise function

Class example # 5 Jump Different values from the left and the right sides Piecewise functions

Class Examples #2 Domain x>1 Continuous for all points greater than 1

Closure What are three types of discontinuity. Give an example of each.

End Behavior of polynomial functions y = p(x) = anxn + an-1 xn-1 + an-2 xn-2 +……. . a 2 x 2 + ax + ao n>0 an positive, n: even y = x 2 y → as x→- y → as x→ an negative, n: even y = - x 2 y → - as x→- y → - as x→ an: positive, n: odd an negative, n: odd y= x 3 y→ - as x→- y→ as x→ y= - x 3 y→ as x→- y → - as x→