Continuity Learning

Let 'a' be an aggregate and a in A, A function f : A |-> R is said to be continuous at "A". If each E > 0 EE   delta >0 such that x in A, || < delta  x - a

rArr   |  f ( x ) - f ( a ) | < E
lim_(x->a) f ( x ) = f ( a )
f is continuous at a    hArr lim_(x->a) f ( x ) = f ( a )

A function f : A |-> Ris said to be continuous on the aggregate 'a'. If ' f ' is continuous at every point a in A
A function f : A -> R is said to be discontinuous at a A. If ' f ' is not continuous at ' a '.


Continuity on the left and right at 'a':-

Let 'a' be an aggregate and a in A. A function f : A|-> R is said to be continuous on the left at 'a'. If each E>0 EE delta >0 such that x in A , a - delta < x <= a  rArr | f ( x ) - f ( a ) | < E

lim_(x->a-)  f ( x ) = f ( a )

Let 'a' be an aggregate and a in A. A function f : A|-> R is said to be continuous on the left at 'a'. If each E>0 EE delta >0 such that x in A , a <= x < a + delta   rArr | f ( x ) - f ( a ) | < E

lim_(x->a+) f ( x ) = f ( a )

Continuity of a function at open and closed intervals:-

A function ' f ' is said to be continuous on an open interval ( a, b ). If ' f ' is continuous at ' x ' AA x in ( a, b ).

A function ' f ' is said to be continuous on an closed interval [ a, b ].

' f ' is continuous at 'x' AA x in ( a, b )
' f ' is right continuous at 'a'.
' f ' is left continuous at ' b'.


Types of discontinuity:-

Let f : A -> Rand a in A be a point of discontinuity of ' f '.

' f ' is said to have Removable discontinuity at 'a'. If lim_(x->a) f ( x ) exists and lim_(x->a) f ( x ) != f ( a ) or f ( a )  is not defined.
' f ' is said to have Jump discontinuity or discontinuity of first kind at 'a'. If lim_(x->a-) f ( x ), lim_(x->a+) f ( x ) both exists and

lim_(x->a-)    f ( x )        !=  lim_(x->a+) f ( x )

' f ' is said to have sample discontinuity at 'a'. If ' f ' has removable discontinuity ( or ) jump discontinuity at 'a'.
' f ' is said to have finite discontinuity at 'a'. If ' f ' is not continuous at 'a' and ' f ' is bounded at 'a'.
' f ' is said to have Infinite discontinuity at 'a'.

If lim_(x->a) f ( x ) = oo   ( or ) lim_(x->a) f ( x ) = -oo
f ( x ) is unbounded in every neighbourhood of 'a'.

Example problems on Continuity

1)Examining the continuity of ' f ' defined by f ( x ) = | x | + | x - 1|

Solution:-    | x| = x if x > 0
= - x if x <= 0
| x - 1| = x if x > 1
= - x if x <= 1

If x <= 0
f ( x ) = | x | + | x - 1|
= - x - ( x -1 )
= 1 - 2x

If 0 < x < 1
f ( x ) =  | x | + | x - 1|
= x  - ( x - 1 )
= x - x + 1
= 1

If x >= 1
f ( x ) = | x | + | x - 1|
= x + x -1
= 2x -1

f ( x ) = 1 - 2x ; If x <= 0
= 1        ;  if 0 < x < 1
= 2x -1  ; if x >= 1

Continuity of ' f ' at x =0

Left Hand Limit            lim_(x->0-) f ( x ) = lim_(x->0-) 1 - 2x = 1

Right Hand Limit          lim_(x->0+) f ( x ) = lim_(x->0+) 1  = 1

f ( x ) = 1 - 2x if x <= 0
f ( 0 ) = 1

lim_(x->0) f ( x ) = f ( 0 ) = 1

Continuity of ' f ' at x =1

Left Hand Limit             lim_(x->1-) f ( x ) = lim_(x->1-) 1  = 1

Right Hand Limit            lim_(x->1+) f ( x ) = lim_(x->1+) 2x - 1  = 1

f ( x ) = 2x -1 if x >= 1
f ( 1 ) = 2 ( 1 )  - 1 = 1

lim_(x->1) f ( x ) = f ( 1 ) = 1

The function is continuous at x = 1