Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous at that point. The function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x and at x 1 We will discuss types of discontinuity later in this section. Section 1.4 Continuity 1
Continuity The following types of functions are continuous over their domain. Polynomials, Rational Functions, Root Functions, Trigonometric Functions, Inverse Trigonometric Functions, Exponential Functions, Logarithmic Functions Theorem: If f and g are continuous at c, a is a real number then each are also continuous at c. i. f + g ii. f g iii. af iv. fg v. f/g, provided gc () 0 Theorem: If g is continuous at c and f is continuous at g(c), then f g is continuous at c. Example 1: Discuss continuity for: a. 5 gt () 5t t Discontinuous: Continuous: x b. f( x) x x6 Discontinuous: Continuous: c. x hx ( ) Discontinuous: Continuous: 1 cosx Section 1.4 Continuity
Types of Discontinuity Removable Discontinuity occurs when: f(c) (or the limit exists, but f(c) is undefined. xc Jump Discontinuity occurs when: and exists, but are not equal. xc xc Infinite Discontinuity occurs when: lim f( x) on at least one side of c. Infinite discontinuities are generally associated xc with having a vertical asymptote. State the type of discontinuity at each point of discontinuity: x = -3 x = 0 x = 1 And so the function is continuous over everywhere else. One-Sided Continuity A function f is called continuous from the left at c if f( c) and continuous from the right at c if f( c). xc In the example above, we have continuity from the right at x = 0 and continuity on the left at x =. Section 1.4 Continuity 3 xc
Continuity Stated a Bit More Formally A function f is said to be continuous at the point x = c if the following three conditions are met: 1. f(c) is defined. exists 3. = f(c) xc To check if a function is continuous at a point, we ll use the three steps above. This process is called the three step method. xc Example : Use the 3 step method to determine continuity or discontinuity. x36, x0 f( x) x 6, x0 We need to check: 1. Is f(0) defined?. Check to see if Must check: x0 exist. x0 and x0 Does exist? x0 3. Does = f(0)? i.e. Compare #1 and # above. x0 If at least one of the three steps fails, identify the type of discontinuity. Section 1.4 Continuity 4
Example 3: Let x1, x f( x) 4 x, x is the function continuous at x =? We need to check: 1. Is f() defined?. Check to see if Must check: x exist. x and x Does exist? x 3. = f()? i.e. Compare #1 and # above. x If at least one of the three steps fails, identify the type of discontinuity. Section 1.4 Continuity 5
Example 4: Is f( x), x3 3 x, 3 x x 9, x 3 continuous at x = 3? We need to check: 1. Is f(3) defined?. Check to see if Must check: exist. x3 x3 and x3 Does exist? x3 3. Does = f(3)? i.e. Compare #1 and # above. x3 If at least one of the three steps fails, identify the type of discontinuity. Section 1.4 Continuity 6
Try this one: Determine if the function f x x We need to check: 1. Is f() defined? ( ) 4 is continuous at x =.. Does exist? Must check x x and. x 3. Does = f()? i.e. Compare #1 and # above. x x 3, x Example 5: Find c so that f(x) is continuous. f( x) cx x, x 1. Find f().. must exist, so make sure that x x =. x 3. Finally set = f() to find c. x Section 1.4 Continuity 7
Try this one: Find A and B so that f(x) is continuous. x 1, x f( x) A, x Bx 3, x 1. Find f(-).. must exist, so make sure that x x =. x 3. Since = f(-) then x x 5 Example 6: The function f( x) is defined everywhere except at x = 5. If possible, x 5 define f ( x ) at 5 so that it becomes continuous at 5. Section 1.4 Continuity 8
Try these: 1 The function f( x) x 3 is defined everywhere except at x = 3. If possible, define f ( x ) at 3 so that it becomes continuous at 3. 1, x x 1, x 4 x Let f( x) x, 4 x6. Find points where the function is discontinuous and classify these 4, x 6 x, x 6 points. Section 1.4 Continuity 9