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

2 Definition a X Y O bC f(c)

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

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

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

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

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

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 X Y O c 2 1

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)

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

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

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

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

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)

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 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 a X Y O b M m Weierstrass theorem

21 aX Y O b M m Weierstrass theorem

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

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

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

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

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

The point c is called a point of discontinuity of the third kind for f(x) in the following case: 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)

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.

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