Liceo Scientifico “Isaac Newton”Roma

Maths course

Continuity

Teacher

Serenella Iacino

XX

YY

OO c

1

f(c)

2

Definition

aa XX

YY

OObbCC

f(c)

3

Definition

• f(x) is defined in c so that f(c) exists

x c

x c

lim f(x) = lim f(x) = ℓx c+x c -

whenf(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) be a function defined in a closed interval [a,b] and let c

be a point belonging to this open interval

XX

f(c)

4

YY

OO c

whenf(x) – f(c) < ε x – c < δ

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

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.

XX

YY

OOc

ℓ

f(x) isn’t continuous at the point c.

L = f(c)

7

if x = c

if x = c

g(x)

L f(x) =

XX

YY

OO c

ℓ

f(x) is continuous at the point c.

8

x clim f(x) = = f(c)ℓ

X

Y

O

= f(c)ℓ

c

f(x) isn’t continuous at the point c.

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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

XX

YY

OO c

ℓ 2

ℓ 1

f(x) isn’t continuous at the point c, but is only right-continuous.

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if x < c

if x > c

g(x)

L

f(x) =

x c +lim f(x) = = lim f(x) =

x c - L ℓ

XX

YY

OO c

L = f(c)

ℓ

if x < c

if x > c

f(x) isn’t continuous at the point c, but is only left-continuous.

if x = c

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g(x)

L f(x) =

h(x)

x c +lim f(x) = = lim f(x) =

x c -L ℓ

XX

YY

OO c

L

ℓ

f(x) isn’t continuous at the point c, but is only right-continuous.

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if x < c

if x > c

if x = c

g(x)

L f(x) =

h(x)

XX

YY

OO c

L

All elementary functions are continuous functions, for example:

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the logarithmic function

the exponential function y = sin x

xx

yy

xx

yy

xx

yy

xx

yy

Parabola

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

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if 0 < x < 3

if 5 < x < 7

x

10-x f(x) =

YY

XXOO 3

3

5

75

Inverse function

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if 0 < x < 3

if 3 < x < 5

x

10-x

f (x) =-1

XX

YY

OO 3

3

5

7

5

lim x = 3 = lim 10 – x = 7+

x 3x 3-

Inverse function

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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

Bolzano theorem

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bb

aa

CC 11 22CC 33CC XX

YY

OO

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.

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aa

XX

YY

OO bb

MM

mm

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

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aa

XX

YY

OO bb

MM

mm

Weierstrass theorem

21

aa XX

YY

OO bb

MM

mm

Weierstrass theorem

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Intermediate value theorem

YY

y = ky = k

aa XXOO

bb

MM

mm

CC 11 CC 22

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

When the function 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:

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

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XX

YY

OO c

ℓ 1

ℓ 2

x c +lim f(x) = and lim f(x) =

x c -ℓ 1 ℓ 2

ℓ 1 ℓ 2jump of f(x) is

“jump discontinuity”

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2. Discontinuity of the second kind

XX

YY

OOc

x c +lim f(x) = + and lim f(x) = -

x c-

∞∞

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XX

YY

OOc

ℓ

2. Discontinuity of the second kind

x c +lim f(x) = - and lim f(x) = ℓ

x c -∞

“ infinite discountinuity”.

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

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3. Discontinuity of the third kind

XX

YY

OO c

ℓ

exists and is x c

lim f(x) = ℓ

finite but the function isn’t defined at the point c

1)

finite but the value of the limit isn’t equal to f(c)

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XX

YY

OO c

ℓ

L = f(c)

exists and is x c

lim f(x) = ℓ 2)

3. Discontinuity of the third kind

“removable discontinuity”.

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