Continuity. Warm-up. Suppose the graph of y = f (x) looks like

Transcription

1 Continuity Warm-up Suppose the graph of y = f x) looks like answers to 4. a x! a x! a What is the domain of f x)?. What is the range of f x)?. For which values a in [, 7] does lim f x) t exist? 4. For those values you picked out in., whatare f x) and lim f x)? + lim 5. Which values a satisfy f a) and lim f x) exist,butf a) 6= lim f x)?

2 Domain definitions Let D be the domain of f x). Ex. D =, ) [, 6) [ 6, 7] Ex. D = { } [, ) [, 6) [ 6, 7] Definition An interior point of D is any point in D which is t an endpoint or an isolated point. Ex. Everything in D except x = 7. Ex. Everything in D except x = & 7. Continuity Let a be an interior point of D Ex. f x) isdiscontinuous at x =4and5. No other points are fair game! Definition A function is continuous at a if lim f x) =f a). If it is t continuous at a, then function is discontinuous at a. Checklist. Is a an interior point? If, stop here... we ll get back to these.. Does a) lim f x) exist?b) lim f x) exist? +. Does lim f x) exist?i.e.doesa) = b)?) 4. Does f a) = lim f x)? If the answer to any of. 4. is, then f x) is discontinuous at a.

3 Some examples Over their domains, all polymials, rational functions, trigometric functions, exponential functions, absolute values, and their inverses are all continuous functions. Jumps all happen over domain gaps) Example Is the function f x) = x x < x + apple x continuous? Solution The only possible problem would happen at x =. Let s check there lim x! f x) = lim x! x = 9 6 lim f x) = lim x += x! + x! + - No, f x) is discontinuous at x = because is an interior point of the domain, but lim x! f x) does t exist.

4 Right Continuity and Left Continuity Definition A function f x) isright continuous at a point a if it is defined on an interval [a, b) andlim + f x) =f a). Similarly, a function f x) isleft continuous at a point a if it is defined on an interval b, a] andlim f x) =f a). Example f x) is a) continuous at every interior point in Dexceptx= 4 and 5; b) only right continuous at those points included in a); and c) additionally left continuous at x =4andx = 7. Suppose a function f has isolated points in its domain. Definition A function f is continuous over its domain D if ) is is continuous at every interior point of D, and) it is left or right) continuous at every endpoint of D. Otherwise,it has a discontinuity at each point in D which violates ) or ). Pick a number a Is a in the domain D? Is a an interior point of D? pick again Is a an endpoint of D? Does the limit exist from the right of a? from the left of a? Is fx) left or right) continuous as appropriate)? Are they equal? Is the two-sided limit equal to fa)? fx) is discontinuous at x=a fx) is continuous at x=a

5 Filling and Fixing Suppose a is a point of discontinuity in D a) If a is an interior point and lim f x) =L exists; or b) if a is an endpoint and lim ± f x) =L exists, then we say f x) hasaremovable discontinuity f x) x 6= a f x) = L x = a Example f x) has a removable discontinuity in exactly one place f x) x 6= 5 f x) = x =5 x * Filling and Fixing Suppose a is a hole in D a is arbitrarily close to points in D, buttind). a) If a would be an interior point and lim f x) =L exists; or b) if a would be an endpoint and lim ± f x) =L exists, then we say f x) hasacontinuous extension f x) x 6= a f x) = L x = a * * x Example f x) has continuous extensions in exactly two places f x) x 6= f x) x 6= f x) = and f x) = x = x =

6 Examples A) Which of the following have removable discontinuities? For those which do, what are the alternate functions with those discontinuities removed? B) Which of the following have continuous extensions? For those which do, what are those extensions?. f x) = x 4 x sin x x 6= /. f x) = 0 x = /. f x) = x x

7 One application The Intermediate Value Theorem Suppose f is continuous on a closed interval [a, b]. If f a) < C < f b) or f a) > C > f b), then there is at least one point c in the interval [a, b] suchthat f c) =C. C C C a c b a b a b Example Show that the equation x 5 one solution in the interval [0, ]. Example Show every polymial x +=0hasatleast px) =a n x n + + a x + a 0, a n 6=0 of odd degree has at least one real root a solution to px) = 0).

8 Where is a function continuous? In general function is continuous at x = a. What does it mean for a function fx) to be continuous at x = a? Explain how to test if a Specifically. For which values of x is the function fx) =x +x + 4 with limits if necessary and draw a graph of the function to illustrate your answer. < x x 6, if x 6=,. For which values of x is the function fx) = x 5, if x =, < sin x, if x 6= 0,. For which values of x is the function fx) = x, if x = 0, cos x 4. For which values of x is the function fx) = x, if x 6= 0,, if x = 0, < sin x, if x 6= 0, 5. Determine the value of k for which the function fx) = 5x is continuous at x = 0. k, if x = 0, Justify your answer x, if apple x<, 6. For which values of x is the function fx) = x, if apple x apple, cos x, if x 0, 7. For which values of x is the function fx) = cos x, if x<0, sin/x), if x 6= 0,. For which values of x is the function fx) = 0, if x = 0,

9 ax +5, if x apple, 9. Find the value of a for which the function fx) = is continuous at x =. Justify x, if x>, your answer +x, if 0 apple x apple, 0. For which values of x is the function fx) = x, if x>,. For which values of x is the function fx) =x x with limits if necessary and draw a graph of the function to illustrate your answer. x, if x<, ><. Find the value of a for which the function fx) = a, if x =, is continuous at x =. Justify > x +, if x>, your answer < x a, if x 6= a,. For which values of x is the function fx) = x a, if x = a, < x x, if x 6= 0, 4. For which values of x is the function fx) =, if x = 0, sin x, if x<0, 5. For which values of x is the function fx) = with x, if x 0, limits if necessary and draw a graph of the function to illustrate your answer. < x n, if x 6=, 6. For which values of x is the function fx) = x n, if x =, 7. Explain how you kw that fx) =secx is continuous for all values of x. Justify your answer with limits if necessary and draw a graph of the function to illustrate your answer.. For which values of x is the function fx) = cos x with limits if necessary and draw a graph of the function to illustrate your answer.

10 9. For which values of x is the function fx) = bxc with limits if necessary and draw a graph of the function to illustrate your answer. x x +x, if x 6=, 0. For which values of x is the function fx) = continuous? Justify your 4, if x =, answer. For which values of x is the function fx) = x + x, apple x apple, Answers. all x. all x. x 6= 0 4. x 6= 0 5. k =/4 6. apple x apple 7. x 6= 0. x 6= 0 9. a = 0.x 0, x6=. all x. a =. x 6= a 4. x 6= 0 5. all x 6. all x 7.. all x 9. x t and integer 0. x 6=. apple x apple