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continuity
speech transcript
Liceo Scientifico Isaac Newton - Roma
continuity
in accordo con il
Ministero dell’Istruzione, Università, Ricerca
e sulla base delle
Politiche Linguistiche della Commissione Europea
percorso formativo a carattere
tematico-linguistico-didattico-metodologico
scuola secondaria di secondo grado
teacher
Serenella Iacino
eniscuola / smartenglish – A cura di Linda Rossi Holden
1
Label europeo 2010
continuity
Continuity
In this video we are going to talk about the continuity of a function.
In mathematics the term continuity indicates one of the most important properties of
functions and we can say intuitively that a function f(x) from the real numbers to real
numbers is continuous if we can draw its graph without taking the pencil off the
paper, which means that its graph is an unbroken curve with no jumps or holes.
A good mathematical definition of continuity was given for the first time in 1821 by
the mathematician Cauchy; his definition, given by the concept of limits, is still used
today, and it’s as follows:
let f(x) be a function defined in a closed interval [a,b] and let c be a point
belonging to this open interval; the function f(x) is said to be continuous at
the point c if:

f(x) is defined in c so that f(c) exists;

lim f(x) exists, is finite and is equal to l;

f(c) = l which means that lim f(x) = f(c)
x
c
x
c
If we use the term neighborhood, let ε be the radius of the neighborhood of f(c) and δ
the radius of the neighborhood of c; so we say that a function f is continuous at the
point c if, for an ε > 0, however small, there exists a δ > 0, such that, for every x in
the neighborhood of c, the value of f(x) satisfies this relation (slide 3).
The following figure represents geometrically the definition of continuity of f(x).
We observe that, while the concept of limit considers the behaviour of the function in
the neighborhood of c irrespective of the image of the function at the point c, the
definition of continuity considers the behaviour of the function either in the
neighborhood of c, or at the point c itself, and the behaviour of each must not be
different.
A function may happen to be continuous in only one direction, either from the left or
from the right.
A function is right – continuous if the
lim f(x) = f(c) likewise it is left –
x
c+
continuous if lim -f(x) = f(c).
x
c
It’s clear that a function is continuous if and only if it’s both right – continuous and left
– continuous, like this one (slide 5).
In addition a function is said to be continuous in the interval [a,b] if it’s continuous at
every point of this interval.
Let us now consider some graphs that illustrate some behaviour of functions at the
point c:
in the graph here shown, the domain of f(x) is the set of the real numbers – {c}, that
is f(c) doesn’t exist; but its limit exists in the right and in the left – neighborhood of c
and is equal to ℓ.
In this case f(x) isn’t continuous at the point c.
In this figure we observe that f(c) exists and is equal to L; in addition both limits
from the right and from the left, as x goes to c, are equal to ℓ but not f(c); so the
function isn’t continuous at the point c.
1
continuity
In this example, f(c) exists; the limits from the right and from left, as x goes to c,
are equal to f(c); so f(x) is continuous at the point c.
In this other case, f(c) doesn’t exist and the limits from the right and from the left
aren’t equal; so f(x) isn’t continuous at the point c.
In this figure f(c) exists and is equal to L; in addition the limits from the right and
from the left aren’t equal; so f(x) isn’t continuous at the point c, but is only right –
continuous.
In this example f(c) exists and is equal to L; in addition the limits from the right and
from the left aren’t equal; so f(x) isn’t continuous at the point c, but is only left –
continuous.
Finally here, f(c) is equal to L; in addition the limit in the left – neighborhood of c
doesn’t exist, while the limit in the right – neighborhood of c exists and is equal to L;
so f(x) isn’t continuous at the point c, but is only right – continuous.
All elementary functions are continuous functions, for example:
all polynomial functions;
the logarithmic function when the argument is greater than 0;
the exponential function;
the trigonometric functions.
In addition if f(x) and g(x) are two continuous functions at the point c, then:
their sum, their product, their quotient, their power and also their composition are still
continuous functions.
Generally, if we want to see for which values of x a function is continuous, we must
study its domain, because the function can’t be continuous at the points in which it
isn’t defined.
We will now consider the continuity of a function and of its inverse function.
If f(x) is a continuous and invertible function in its domain, it doesn’t mean that its
inverse function is continuous.
For example we consider the function f(x) defined thus (slide 15).
This function is continuous and invertible in its domain that is the union of two
intervals; its inverse function is as follows (slide 15).
The inverse function isn’t continuous at x = 3 because f(3) = 3 and the right and the
left limits in this point aren’t equal.
So, given a continuous and invertible function f(x), the continuity of its inverse
function is established by the Inverse function Theorem whose definition is as
follows (slide 17).
Now, we can enunciate three important theorems regarding the continuity of a
function f(x) in a closed interval [a,b]; the first is called Bolzano Theorem and its
definition is as follows (slide 18).
This theorem states that if f(x) is a continuous function in [a,b], and f(a) and f(b)
differ in sign, its graph must have some point of intersection with the x axis.
Now let’s see another theorem called Weierstrass Theorem whose definition is as
follows (slide 19).
In these hypotheses, the function in figure attains its minimum in a point inside the
closed interval [a,b] and its Maximum in the extreme b; or inside the closed interval
[a,b]; or in the extremes of the closed interval [a,b].
Finally, let’s see the third theorem called Intermediate value Theorem whose
definition is as follows (slide 22).
2
continuity
This theorem states that, in these hypotheses, each straight – line, having equation
y = K, with K limited between m and M, meets the function f(x) in some real point.
When the function f(x) isn’t continuous at the point c, we can say that f(x) has a
discontinuity at that point.
We can then distinguish three types of different discontinuities as follows:
The point c is called a point of discontinuity of the first kind for f(x), if the limits from
the right and from the left, as x goes to c, exist, are finite, but aren’t equal, as we can
see in the figure.
This type of discontinuity is also called a jump discontinuity and the number
l L - L 2 l is called the jump of the function.
1
In this case the point c is called a point of discontinuity of the second kind for f(x), if
at least one of the limits from the right or from the left, doesn’t exist or is equal to
infinity, as we can see in the figure.
This graph is another example of discontinuity of the second kind; here there’s only
the limit from the right; this type of discontinuity is also called an infinite
discontinuity.
The point c is called a point of discontinuity of the third kind for f(x) in the following
case: the limit exists and is finite but the function isn’t defined at the point c;
or in the following case: here, the limit exists and is finite but it isn’t equal to f(c);
this type of discontinuity is also called removable discontinuity.
And this is all for today. Enjoy learning maths!
Materiale sviluppato da eniscuola nell’ambito del protocollo d’intesa con il MIUR
3
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