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Unformatted text preview: to know the behaviour of a function near any chosen instant
a : a little to the left of a or (a − δx) and a little to the right of a or (a + δx).
left
right
39 Geometrically
Geometrically when we say that a SINGLE VALUED function is CONTINUOUS
around the point or instant a we mean that the curve of the function is without
a,
breaks or gaps.
f(x)
3 value f(a)=3 f(x) Limit +f(x) = 3
x→a 2
1 (0,0
0,0)
( 0,0 ) Limit  f(x) =3
x→a a  δ x a a +δ x x f(x) is CONTINUOUS at the point or instant a f(x)
3.5
3 Limit  f(x) =3
x→a value f(a)=3 Limit +f(x) = 3.5
x→a 2
1 (0 , 0) a  δ x a a +δ x x f(x) is CONTINUOUS at instant a bu t only from the left
l eft f(x) value f(a)=3.5 Limit  f(x) =3
x→a 3.5
3 Limit +f(x) = 3.5
x→a 2
1 0,0)
(0, 0) a  δ x a a +δ x x f(x) is CONTINUOUS at instant a bu t only from the right
r ight 40 Analytically : Assume Value f(x) = f(a) exists and is something definite. Then
x=a 1. if xL imit  f(x) exists and is = Value f(x)
→a
xa
= we say that f(x) is CONTINUOUS at the point x = a from the left .
l eft
2. if L imit+ f(x) exists and is = Value f(x)
x→a x=a we say that f(x) is CONTINUOUS at the point x = a from the right .
r ight
f(x) is said to be CONTINUOUS at the point or instant x = a
point instant
if L imit f(x) = f(a) = L imit f(x)
x → a  x → a+ Using the infinitesimal δ x we can write this as:
f(x) is said to be CONTINUOUS at the point or instant x = a
if Limit f(a  δ x) = f(a) = Limit f(a + δ x)
δx → 0 δx → 0 If f(x) is not CONTINUOUS at a we say f(x) is DISCONTINUOUS at a.
We may test the continuity of a function using the decision tree below.
continuity
contin
f(x) CONTINUOUS at a ? LIMIT x→a ?
= ALUE
VALUE f(x) = f(a)
x=a directly evaluate with x = a LIMIT LIMIT ?
= LIMIT
x→a
x → a+
irst
take
simplify ffir st then tak e the limit
ir
41 polynomials
Are the pol ynomials C ONTINUOUS ever ywhere ?
p ol
We can easily see that f(x) = xn for n = 1, 2, 3, . . . Limit f(x) = Value f(x) = an
x→a
x =a
Since a can be any point on the real number line, these f(x) are continuous
continuous
everywhere.
Likewise, f(x) = an xn for n = 0, 1, 2, 3, . . . continuous
where the coefficients an are real numbers, are also continuous everywhere.
contin
Combining both we get the general polynomial f(x) :
f(x) = a n x n + a n 1 x n1 + . . . + a 2 x 2 + a 1 x + a 0
These are single valued and continuous everywhere.
single valued
continuous
sing
contin
y(t) = u . s i n θ . t  1 g t 2 is a polynom...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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