Section 2.8 - Continuity 2.2

A function f(x) is continuous at x = c if and only if all three of the following tests hold: f(x) is right continuous at x = -5 f(x) is continuous at x = -4 f(x) has infinite discontinuity at x = -3 [i, iii] f(x) has point discontinuity at x = -2 [i, iii] f(x) has infinite discontinuity at x = -1 [i, ii, iii] f(x) is continuous at x = 0

Point Discontinuity [i, iii] At x = 1 At x = 2 At x = 3 At x = 4 At x = 5 Point Discontinuity [i, iii] Jump Discontinuity [i, ii, iii] Continuous Continuous Point Discontinuity [i, (ii), iii]

continuous continuous pt. discontinuity at x = 0 inf. discontinuity at x = 1 pt. discontinuity at x = 3 continuous inf. discontinuity at x = -3 jump discontinuity at x = 2

Find the value of a which makes the function below continuous

Find (a, b) which makes the function below continuous As we approach x = -1 2 = -a + b As we approach x = 3 -2 = 3a + b

Consider the function Find the value of k which makes f(x) continuous at x = 0 Since , if k =1, the hole is filled.

Calculator Required Let m and b be real numbers and let the function f be defined by: If f is both continuous and differentiable at x = 1, then:

No Calculator The function f is continuous at x = 1

Calculator Required Which of the following is true about I. f is continuous at x = 1 II. The graph of f has a vertical asymptote at x = 1 III. The graph of f has a horizontal asymptote at y = 1/2 I. f(1) results in zero in denominator….NO II. Since x – 1 results in 0/0, it is a HOLE, NOT asymptote X X III. X X

No Calculator Which function is NOT continuous everywhere? undefined at x = -1

Calculator Required The graph of the derivative of a function f is shown below. Which of the following is true about the function f? I. f is increasing on the interval (-2, 1) II. f is continuous at x = 0 III. f has an inflection point at x = 2 NO YES NO A. I B. II C. III D. II, III E. I, II, III

No Calculator A. 2 B. 1 C. 0 D. -1 E. -2