Download presentation

Presentation is loading. Please wait.

Published byEthan White Modified over 4 years ago

1
Liceo Scientifico Isaac Newton Roma Maths course Continuity Teacher Serenella Iacino X Y O c 1 f(c)

2
2 Definition a X Y O bC f(c)

3
3 Definition f(x) is defined in c so that f(c) exists x c lim f(x) = x c + - whenf(x) – f(c)< εx – c< δ lim f(x) exists, is finite and is equal to so that f(c)= which means that lim f(x) = f(c) Let f(x) be a function defined in a closed interval [a,b] and let c be a point belonging to this open interval

4
X f(c) 4 Y O c whenf(x) – f(c)< εx – c< δ

5
5 lim f(x) = f(c) x c - lim f(x) = f(c) x c + + lim f(x) = lim f(x) = f(c) x c - right-continuous left-continuous

6
6 f(c) doesnt exist x c + lim f(x) = x c - f(x) isnt continuos at the point c. X Y O c

7
f(x) isnt continuous at the point c. L = f(c) 7 if x = c g(x) L f(x) = X Y O c

8
f(x) is continuous at the point c. 8 x c lim f(x) = = f(c) X Y O = f(c) c

9
f(x) isnt continuous at the point c. 9 if x < c if x > c f(x) = 1 2 x c + lim f(x) = = lim f(x) = x c - 1 2 X Y O c 2 1

10
f(x) isnt continuous at the point c, but is only right-continuous. 10 if x < c if x > c g(x) L f(x) = x c + lim f(x) = = lim f(x) = x c - L X Y O c L = f(c)

11
if x < c if x > c f(x) isnt continuous at the point c, but is only left-continuous. if x = c 11 g(x) L f(x) = h(x) x c + lim f(x) = = lim f(x) = x c - LX Y O c L

12
f(x) isnt continuous at the point c, but is only right-continuous. 12 if x < c if x > c if x = c g(x) L f(x) = h(x) X Y O c L

13
All elementary functions are continuous functions, for example: 13 the logarithmic function the exponential functiony = sin x x y x y x y x y Parabola

14
14 f(x) + g(x) f(x) g(x) f(x) g(x) [f(x)] g(x) is still continuous In addition, if f(x) and g(x) are two continuous functions at the point c, then: f [ g (x) ]is still continuous

15
15 if 0 < x < 3 if 5 < x < 7 x 10-x f(x) = Y XO 3 3 5 7 5 Inverse function

16
16 if 0 < x < 3 if 3 < x < 5 x 10-x f (x) = X Y O 3 3 5 7 5 lim x = 3 = lim 10 – x = 7 + x 3 - Inverse function

17
17 Inverse function theorem Let I be a limited or unlimited interval and let f(x) be a function defined in I and here continuous. If f(x) is invertible then is continuous. f (x)

18
Bolzano theorem 18 b aC1 2C 3CX Y O Let f(x) be a function defined and continuous in a closed and limited interval [a, b]. If f(a) f(b) < 0 then theres a point c belonging to the open interval (a, b) such that f(c) = 0.

19
19 a X Y O b M m Let f(x) be a function defined and continuous in a closed interval [a, b]; then the function attains its Maximum and its minimum in [a, b]; so theres at least a point c belonging to this interval such that: f(x) f(c) or f(x) f(c) for all x belonging to the closed interval [a, b]. Weierstrass theorem

20
20 a X Y O b M m Weierstrass theorem

21
21 aX Y O b M m Weierstrass theorem

22
22 Intermediate value theorem Y y = k a X O b M mC1C2 Let f(x) be a continuous function in a closed and limited interval [a, b]; if m and M are its minimum and Maximum values in this interval, and if K is a number between m and M, then theres some number c in [a, b] such that f(c)=K

23
When the function f(x) isnt continuous at the point c, we say that f(x) has a discontinuity at that point. We can then distinguish three types of different discontinuities as follows: 1.Discontinuity of the first kind 2. Discontinuity of the second kind 3. Discontinuity of the third kind Discontinuity

24
1.Discontinuity of the first kind 24 X Y O c 1 2 x c + lim f(x) = and lim f(x) = x c - 1 2 1 2 jump of f(x) is jump discontinuity

25
25 2. Discontinuity of the second kind X Y O c x c + lim f(x) = + and lim f(x) = - x c -

26
26 X Y O c 2. Discontinuity of the second kind x c + lim f(x) = - and lim f(x) = x c - infinite discountinuity.

27
The point c is called a point of discontinuity of the third kind for f(x) in the following case: 27 3. Discontinuity of the third kind X Y O c exists and is x c lim f(x) = finite but the function isnt defined at the point c 1)

28
finite but the value of the limit isnt equal to f(c) 28 X Y O c L = f(c) exists and is x c lim f(x) = 2) 3. Discontinuity of the third kind removable discontinuity.

29
29 Copyright 2012 © eni S.p.A.

Similar presentations

OK

Lets play bingo!!. Calculate: MEAN 4 7 5 8 6 Calculate: MEDIAN 9 4 6 1 5.

Lets play bingo!!. Calculate: MEAN 4 7 5 8 6 Calculate: MEDIAN 9 4 6 1 5.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on prepositions for grade 6 Ppt on depth first search complexity Maths ppt on binomial theorem Ppt on ms powerpoint tutorial Ppt on natural and artificial satellites of india Ppt on communist party of india Ppt on asian continents Ppt on computer graphics and virtual reality Ppt on mahatma gandhi as a leader Ppt on domain and range