THE CONTINUITY_26 anno 2003 PPT DEFINITIVO SENZA NOTE[1]

LICEO SCIENTIFICO ISAAC NEWTON - ROMA Maths course CONTINUITY by Professor Iacino Serenella : 

LICEO SCIENTIFICO ISAAC NEWTON - ROMA Maths course CONTINUITY by Professor Iacino Serenella X Y O c 1 f(c)

PowerPoint Presentation: 

2 DEFINITION a X Y O b C f(c)

PowerPoint Presentation: 

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

PowerPoint Presentation: 

X f(c) 4 Y O c when f(x) – f(c) < ε x – c < δ

PowerPoint Presentation: 

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

PowerPoint Presentation: 

6 f(c) doesn’t exist x c + lim f(x) = lim f(x) = x c - ℓ f(x) isn’t continuos at the point c . X Y O c ℓ

PowerPoint Presentation: 

f(x) isn’t continuous at the point c . L = f(c) 7 if x = c if x = c g(x) L f(x) = X Y O c ℓ

PowerPoint Presentation: 

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

PowerPoint Presentation: 

f(x) isn’t continuous at the point c . 9 if x < c if x > c f(x) = ℓ 1 ℓ 2 x c + lim f(x) = = lim f(x) = x c - ℓ 1 ℓ 2 X Y O c ℓ 2 ℓ 1

PowerPoint Presentation: 

f(x) isn’t 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 ℓ X Y O c L = f(c) ℓ

PowerPoint Presentation: 

if x < c if x > c f(x) isn’t 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 - L ℓ X Y O c L ℓ

PowerPoint Presentation: 

f(x) isn’t 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

PowerPoint Presentation: 

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

PowerPoint Presentation: 

14 f(x) + g(x) f(x) ● g(x) f(x) g(x) [f(x)] g(x) is still continuous is still continuous is still continuous 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

PowerPoint Presentation: 

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

PowerPoint Presentation: 

16 if 0 < x < 3 if 3 < x < 5 x 10-x f (x) = -1 X Y O 3 3 5 7 5 lim x = 3 = lim 10 – x = 7 + x 3 x 3 - Inverse function

PowerPoint Presentation: 

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) -1

PowerPoint Presentation: 

Bolzano theorem 18 b a C 1 2 C 3 C X 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 there’s a point c belonging to the open interval (a , b) such that f(c) = 0.

PowerPoint Presentation: 

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 there’s 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

PowerPoint Presentation: 

20 a X Y O b M m Weierstrass theorem

PowerPoint Presentation: 

21 a X Y O b M m Weierstrass theorem

PowerPoint Presentation: 

22 Intermediate value theorem Y y = k a X O b M m C 1 C 2 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 there’s some number c in [a , b] such that f(c)=K

PowerPoint Presentation: 

When the fuction f(x) isn’t 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: DISCONTINUITY OF THE FIRST KIND 2. DISCONTINUITY OF THE SECOND KIND 3. DISCONTINUITY OF THE THIRD KIND DISCONTINUITY

PowerPoint Presentation: 

DISCONTINUITY OF THE FIRST KIND 24 X Y O c ℓ 1 ℓ 2 x c + lim f(x) = and lim f(x) = x c - ℓ 1 ℓ 2 ℓ 1 ℓ 2 jump of f(x) is “ jump discontinuity”

PowerPoint Presentation: 

25 2. DISCONTINUITY OF THE SECOND KIND X Y O c x c + lim f(x) = + and lim f(x) = - x c - ∞ ∞

PowerPoint Presentation: 

26 X Y O c ℓ 2. DISCONTINUITY OF THE SECOND KIND x c + lim f(x) = - and lim f(x) = ℓ x c - ∞ “ infinite discountinuity”.

PowerPoint Presentation: 

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 ℓ exists and is x c lim f(x) = ℓ finite but the function isn’t defined at the point c 1)

PowerPoint Presentation: 

finite but the value of the limit isn’t equal to f(c) 28 X Y O c ℓ L = f(c) exists and is x c lim f(x) = ℓ 2) 3. DISCONTINUITY OF THE THIRD KIND “ removable discontinuity”.

PowerPoint Presentation: 

THE END 29