Continuity Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / - PowerPoint PPT Presentation

Continuity Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 10

Section 3.2 :: Continuity 2 / 10

Continuity at x = c A function f is continuous at x = c if the following three conditions are all satisfied: 1 f ( c ) is defined, 2 lim x → c f ( x ) exists, and 3 lim x → c = f ( c ). If f is not continuous at c , then we say it is discontinuous there. 3 / 10

Identifying Discontinuities 6 4 2 − 3 − 2 − 1 1 2 3 − 2 − 4 4 / 10

Identifying Discontinuities 10 8 6 4 2 − 8 − 6 − 4 − 2 2 4 6 8 − 2 − 4 − 6 − 8 5 / 10

Identifying Discontinuities 6 4 2 − 8 − 6 − 4 − 2 2 4 6 8 − 2 − 4 − 6 6 / 10

Types of Functions and their Continuity Properties • Polynomial Functions continuous everywhere • Rational Functions continuous wherever defined • Square Root Functions continuous where radicand is non-negative • Exponential Functions continuous everywhere • Logarithmic Functions continuous on interval of positive real numbers 7 / 10

Continuity on a Closed Interval A function is continuous on a closed interval [ a , b ] if 1 it is continuous on the open interval ( a , b ), 2 it is continuous from the right at x = a , and 3 it is continuous from the left at x = b . 8 / 10

Continuity of Piecewise-Defined Functions Consider the function  x − 1 , x < 1   f ( x ) = 0 , 1 ≤ x ≤ 4  x − 2 , x > 4 .  Where is f ( x ) continuous? 9 / 10

Continuity of Piecewise-Defined Functions Consider the function x 3 + k , � x ≤ 3 f ( x ) = kx − 5 , x > 3 . Find the value of k so that f ( x ) is continuous at x = 3. 10 / 10