Continuity of Functions in Mathematics
Explore the concept of continuity of functions, including definitions, tests, and examples. Learn how to determine if a function is continuous at a point using specific criteria. Dive into the application of continuity tests through real examples.
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Presentation Transcript
Continuity of a function Definition Definition Continuity Test Continuity Test Properties of Continuous Properties of Continuous Functions Continuous Extension to a Point Continuous Extension to a Point Functions
Continuity of a function f(x) Any function y= f(x) whose graph can be sketched over its domain in one continuous motion without lifting the pencil is an example of a continuous function. In this lecture we investigate more precisely what it means for a function to be continuous.
Definition Definition Continuous Continuous at at a a Point Point Interior Interior point point: : A A function function y= y= f(x) f(x) is is continuous continuous at at an an interior interior point point c c of of its its domain domain if if lim ? ?? ? = ?(?) Endpoint Endpoint: : A A function function y= y= f(x) f(x) is is continuous continuous at at a a left left endpoint endpoint a a or or is is continuous continuous at at a a right right endpoint endpoint b b of of its its domain domain if if ? ?+? ? = ?(?) or or lim ? ? ? ? = ?(?) respectively respectively. . lim
Continuity Test Continuity Test A function ( A function (x x) is continuous at ) is continuous at x=c x=c if and only if it meets the following three if and only if it meets the following three conditions. conditions. 1. ( 1. (c c) exists ) exists ( (c c lies in the domain of ) lies in the domain of ) 2. 2. ??? ? ?? ? exists exists ( has a limit as ( has a limit as ? ?) ) 3. 3. ??? ? ?? ? = ?(?) (the limit equals the function value) (the limit equals the function value)
Example: Example: Apply the continuity test to check whether the following functions are continuous at x=0 or not.
Sol Sol. . The function at graph (a) is continuous because ? 0 ? ? =1 1. lim ? 0+? ? = lim 2. f (0) = 1 3. lim ? 0? ? = ?(0) The function at graph (b) has a limit at x=0 ? 0 ? ? =1 , but it is undefined at x=0 then it is discontinuous. lim ? 0+? ? = lim The function at graph (c) has a limit at x=0 ? 0 ? ? =1 , but f (0 )=2 lim ? 0+? ? = lim Since ?(0) lim ? 0? ? then it is discontinuous.
Theorem Theorem (a) Every polynomial ? ? = ????+ ?? 1?? 1+ ?? 2?? 2+...+?0 is continuous because lim ? ?? ? = ?(?) (b) If P(x) and Q(x) are polynomials then the rational function P(x) / Q(x) is continuous wherever it is defined provided that ?(?) 0 (c) All trigonometric functions wherever they are defined.
Properties of Continuous Functions Properties of Continuous Functions If the functions and g are continuous at x=c then the following combinations are continuous at 1. 1. Sums: f + g f + g 2. 2. Differences: f f - - g g 3. 3. Products: f . g f . g 4. 4. Constant multiples: k . f k . f for any number k 5. 5. Quotients: f / g f / g provided that ?(?) 0 6. 6. Powers: ? ? ? provided it is defined on an open interval containing c, where r and s are integers.
Example Example Show that the following functions are continuous everywhere on their respective domains ? 2 3 1+?4 ? 2 ?2 2 ?2 2? 5 ? ? = ? ? = ? ? = Sol. Sol. (a) The square root function is continuous on [0, ) because it is a rational power of the continuous identity function (f(x)=x). The given function is then the composite of the polynomial ? ? = ?2 2? 5 with the square root function ? ? = ?.
Sol. Sol. (b) The numerator is a rational power of the identity function; the denominator is an everywhere-positive polynomial. Therefore, the quotient is continuous. Sol. Sol. (c) The quotient is continuous for all ? 2 and the function is the composition of this quotient with the continuous absolute value function.
