1 / 28

Continuity

Continuity. Alex Karassev. Definition. A function f is continuous at a number a if Thus, we can use direct substitution to compute the limit of function that is continuous at a. Some remarks. Definition of continuity requires three things: f(a) is defined (i.e. a is in the domain of f)

sherriray
Télécharger la présentation

Continuity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Continuity Alex Karassev

  2. Definition • A function f is continuous at a number a if • Thus, we can use direct substitution to compute the limit of function that is continuous at a

  3. Some remarks • Definition of continuity requires three things: • f(a) is defined (i.e. a is in the domain of f) • exists • Limit is equal to the value of the function • The graph of a continuous functions does not have any "gaps" or "jumps"

  4. Continuous functions and limits • TheoremSuppose that f is continuous at band Then • Example

  5. Properties of continuous functions • Suppose f and g are both continuous at a • Then f + g, f – g, fg are continuous at a • If, in addition, g(a) ≠ 0 then f/g is also continuous at a • Suppose that g is continuous at a and f is continuous at g(a). Then f(g(x)) is continuous at a.

  6. Which functions are continuous? • Theorem • Polynomials, rational functions, root functions, power functions, trigonometric functions, exponential functions, logarithmic functions are continuous on their domains • All functions that can be obtained from the functions listed above using addition, subtraction, multiplication, division, and composition, are also continuous on their domains

  7. Example • Determine, where is the following function continuous:

  8. Solution • According to the previous theorem, we need to find domain of f • Conditions on x: x – 1 ≥ 0 and 2 – x >0 • Therefore x ≥ 1 and 2 > x • So 1 ≤ x < 2 • Thus f is continuous on [1,2)

  9. Intermediate Value Theorem

  10. River and Road

  11. River and Road

  12. Definitions • A solution of equation is also calledarootof equation • A number c such that f(c)=0 is calledarootof function f

  13. Intermediate Value Theorem (IVT) • f is continuous on [a,b] • N is a number between f(a) and f(b) • i.e f(a) ≤ N ≤ f(b) or f(b) ≤ N ≤ f(a) • then there exists at least one c in [a,b] s.t. f(c) = N y y = f(x) f(b) N f(a) x c a b

  14. Intermediate Value Theorem (IVT) • f is continuous on [a,b] • N is a number between f(a) and f(b) • i.e f(a) ≤ N ≤ f(b) or f(b) ≤ N ≤ f(a) • then there exists at least one c in [a,b] s.t. f(c) = N y y = f(x) f(b) N f(a) x c3 c1 c2 a b

  15. Equivalent statement of IVT • f is continuous on [a,b] • N is a number between f(a) and f(b), i.e f(a) ≤ N ≤ f(b) or f(b) ≤ N ≤ f(a) • then f(a) – N ≤ N – N ≤ f(b) – N or f(b) – N ≤ N – N ≤ f(a) – N • so f(a) – N ≤ 0 ≤ f(b) – N or f(b) – N ≤ 0 ≤ f(a) – N • Instead of f(x) we can consider g(x) = f(x) – N • so g(a) ≤ 0 ≤ g(b) or g(b) ≤ 0 ≤ g(a) • There exists at least one c in [a,b] such that g(c) = 0

  16. Equivalent statement of IVT • f is continuous on [a,b] • f(a) and f(b) have opposite signs • i.e f(a) ≤ 0 ≤ f(b) or f(b) ≤ 0 ≤ f(a) • then there exists at least one c in [a,b] s.t. f(c) = 0 y y = f(x) f(b) c x a N = 0 b f(a)

  17. y 1 x -1 0 1 -1 Continuity is important! • Let f(x) = 1/x • Let a = -1 and b = 1 • f(-1) = -1, f(1) = 1 • However, there is no c such that f(c) = 1/c =0

  18. Important remarks • IVT can be used to prove existence of a root of equation • It cannot be used to find exact value of the root!

  19. Example 1 • Prove that equation x = 3 – x5 has a solution (root) • Remarks • Do not try to solve the equation! (it is impossible to find exact solution) • Use IVTto prove that solution exists

  20. Steps to prove that x = 3 – x5 has a solution • Write equation in the form f(x) = 0 • x5 + x – 3 = 0 so f(x) = x5 + x – 3 • Check that the condition of IVT is satisfied, i.e. that f(x) is continuous • f(x) = x5 + x – 3 is a polynomial, so it is continuous on (-∞, ∞) • Find a and b such that f(a) and f(b) are of opposite signs, i.e. show that f(x) changes sign (hint: try some integers or some numbers at which it is easy to compute f) • Try a=0: f(0) = 05 + 0 – 3 = -3 < 0 • Now we need to find b such that f(b) >0 • Try b=1: f(1) = 15 + 1 – 3 = -1 < 0 does not work • Try b=2: f(2) = 25 + 2 – 3 =31 >0 works! • Use IVT to show that root exists in [a,b] • So a = 0, b = 2, f(0) <0, f(2) >0 and therefore there exists c in [0,2] such that f(c)=0, which means that the equation has a solution

  21. x = 3 – x5⇔ x5 + x – 3 = 0 y 31 x 0 2 N = 0 c (root) -3

  22. Example 2 • Find approximate solution of the equationx = 3 – x5

  23. Idea: method of bisections • Use the IVT to find an interval [a,b] that contains a root • Find the midpoint of an interval that contains root: midpoint = m = (a+b)/2 • Compute the value of the function in the midpoint • If f(a) and f (m) are of opposite signs, switch to [a,m] (since it contains root by the IVT),otherwise switch to [m,b] • Repeat the procedure until the length of interval is sufficiently small

  24. f(x) = x5 + x – 3 = 0 We already know that [0,2] contains root f(x)≈ > 0 < 0 31 -3 -1 Midpoint = (0+2)/2 = 1 0 2 x

  25. f(x) = x5 + x – 3 = 0 f(x)≈ 31 6.1 -3 -1 1.5 0 2 1 x Midpoint = (1+2)/2 = 1.5

  26. f(x) = x5 + x – 3 = 0 f(x)≈ 31 6.1 -3 1.3 -1 0 2 1.5 1 1.25 x Midpoint = (1+1.5)/2 = 1.25

  27. f(x)≈ f(x) = x5 + x – 3 = 0 31 6.1 -3 1.3 -.07 -1 1.25 1.125 1 0 2 1.5 Midpoint = (1 + 1.25)/2 = 1.125 x • By the IVT, interval [1.125, 1.25] contains root • Length of the interval: 1.25 – 1.125 = 0.125 = 2 / 16 = = the length of the original interval / 24 • 24 appears since we divided 4 times • Both 1.25 and 1.125 are within 0.125 from the root! • Since f(1.125) ≈ -.07, choose c ≈ 1.125 • Computer gives c ≈ 1.13299617282...

  28. Exercise • Prove that the equationsin x = 1 – x2has at least two solutions Hint: Write the equation in the form f(x) = 0 and find three numbers x1, x2, x3, such that f(x1) and f(x2) have opposite signs AND f(x2) and f(x3) haveopposite signs. Then by the IVT the interval [ x1, x2 ] contains a root ANDthe interval [ x2, x3 ] contains a root.

More Related