Lesson 2-5

1. Lesson 2-5 Continuity

2. Objectives • Understand and use the definition of continuity • Understand and use the Intermediate Value Theorem

3. Vocabulary • Continuity – no gaps in the curve (layman’s definition) • Discontinuity – a point where the function is not continuous • Removable discontinuity – a discontinuity that can be removed by redefining the function at a point also called a point discontinuity • Infinite discontinuity – a discontinuity because the function increases or decreases without bound at a point • Jump discontinuity – a discontinuity because the function jumps from one value to another • Continuous from the right at a number a – the limit of f(x) as x approaches a from the right is f(a) • Continuous from the left at a number a – the limit of f(x) as x approaches a from the left is f(a) • A function is continuous on an interval if it is continuous at every number in the interval

4. 1/x if x ≠ 0 f(x) = 0 if x = 0 x²/x if x ≠ 0 f(x) = 1 if x = 0 Continuity • Definition: A function is continuous at a number a if • Note: that the definition implicitly requires three things of the function • f(a) is defined (i.e., a is in the domain of f) • f has a discontinuity at a, if f is not continuous at a. Note the graphs of the examples of discontinuities below: lim f(x) = f(a) xa lim f(x) exisits xa lim f(x) = f(a) xa Removable Infinite Removable Jump x² - x - 2 f(x) = -------------- x - 2 f(x) = [[x]]

5. Continuity Theorems lim f(x) = f(a) xa • If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: • f + g • f – g • cf • f • g • f / g if g(a) ≠ 0 • Any polynomial is continuous everywhere; that is, its continuous on (-∞,∞) • The following types of functions are continuous at every number in their domains: polynomials rational functions trigonometric functions inverse trigonometric functions exponential functions logarithmic functions root functions

6. Continuity Examples • |x| • f(x) = ------ is f(x) continuous at x = 1? x • f(x) = x² + 1 is f(x) continuous at x = 0? |x| / x for x ≠ 0 • f(x) = is f(x) continuous at x = 0? 0 for x = 0

7. Continuity Examples (cont) • x² - 4 --------- x ≠ 2 x - 2 • f(x) = is f(x) continuous at x = 2? 4 x = 2 1/x x ≤ -1 is f(x) continuous at x = -1? • f(x) = (x-1)/2 -1 < x < 1 is f(x) continuous at x = 1?x x ≥ 1

8. Continuity Examples (cont) Is f(x) =  x² + 6x + 9 continuous for all real numbers?

9. Continuity from a Graph a) At which points is the graph discontinuous? b) On what intervals is the graph continuous?

10. Intermediate Value Theorem (IVT) If f is continuous on [a, b] and if W is a number between f(a) and f(b), then there is a number c between a and b such that f(c) = W. IVT is an existence theorem. This means there can be more than one value in [a,b] that will satisfy the theorem! (like in the graph) f(b) W f(a) a b c c Example: You stop at a light. Twenty-five seconds down the street you look at your speedometer and it reads 35 mph. Sometime between the light and then your speedometer had to read 25 mph!

11. IVT Example Show that f(x) = x3+ 2x - 1 has a zero on the interval [0, 1]. f(x) is a polynomial and is continuous everywhere. f(0) = -1 and f(1) = 2 Since f(x) is continuous on [0,1], then the IVT applies and there must be a c between a and b such that f(c) = 0

12. Summary & Homework • Summary: • Formal Definition of a limit requires finding an ε and δ (margins of errors around a and f(a) respectively) • Homework: pg 133-135: 7, 10, 15-18, 35, 40, 43