Continuity And Differentiability :
Bernard Bolzano first given the definition of continuity in 1817, after this Augustin-Louis Cauchy Defined continuity as an infinitely small increment of the independent variable
x
in
y
=
f
(
x
) always leads to infinitely small change in dependent variable
y
. Here infinite small quantities were defined in terms of variable quantities and this definition of continuity closely parallels the infinitesimal definition used today.
Bolzano later in 1830 gave the formal definition and the distinction between pointwise continuity and uniform continuity but his work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point
c
unless it was defined at and on both sides of
c
, but Edouard Goursat allowed the function to be defined only at and on one side of
c
, and later Camille Jordan allowed it even if the function was defined only at
c
. All three of those non-equivalent definitions of pointwise continuity are still in use.
Now we can define continuity formally as
which can be further elaborated as
A function is said to be differentiable if its derivative exists at each point of its domain or having non vertical tangent line at each point of its domain, Graphs of Differentiable functions are always smooth and do not contain any break. Differentiability of a function can be defined as
. So, if a function is differentiable then it must hold the equality as well as LHS and RHS must be defined.
Continuity And Differentiability Introduction
Continuity of a function is defined as
here
means limiting value of function at a and f(a) represents value of function at point a.
Or
means Right Hand Limit or moving towards a from right side of
a
.
means Left Hand Limit or moving towards
a
from left hand side of
a
Examples based on continuity of functions
Example 1: -
Check Continuity of function
f
(
x
) =
x
2
at
x
= 2?
Sol.
For continuity of function,
must hold.
f
(2) = 2
2
= 4 and now
can be written further
means moving towards 2 from the right-hand side.
so,
x
can take values as 2.1,2.01,2.001 so on. as it is approaching towards 2. Now
at
x
= 2.1,
f
(
x
) = (2.1)
2
= 4.41
at
x
= 2.01,
f
(
x
) = (2.01)
2
= 4.0401
at
x
= 2.001,
f
(
x
) = (2.001)
2
= 4.004001
from above we can say as x approaches towards 2 values of function approaching towards 4
hence,
means towards 2 from the left-hand side.
So,
x
can take values as 1.9,1.99,1.999 so on.as it is approaching 2.
at
x
= 1.9,
f
(
x
) = (1.9)
2
= 3.61
at
x
= 1.99,
f
(
x
) = (1.99)
2
= 3.9601
at
x
=1.999,
f
(
x
) = (1.999)
2
= 3.996001
from above we can say as
x
approaches towards 2 values of function approaching towards 4 hence
hence function is continuous at 2.
Example 2: -
Check continuity of function
f
(
x
) = [
x
]
2
at
x
= 3?
Sol.
Here
must hold if function is continuous at 3,
f
(3) = 3
2
= 9
,
Now for
we can take values as 3.1, 3.01, 3.001 so on as shown in graph below
at
x
= 3.1,
f
(
x
) = [3.1]
2
= 3
2
at
x
= 3.01,
f
(
x
) = [3.01]
2
= 3
2
at
x
= 3.001,
f
(
x
) = [3.001]
2
= 3
2
hence
For
we can put values of x as 2.9, 2.99, 2.999 so on as shown in graph below
at
x
= 2.9,
f
(
x
) = [2.9]
2
= 2
2
at
x
= 2.99,
f
(
x
) = [2.99]
2
= 2
2
at
x
= 2.999,
f
(
x
) = [2.999]
2
= 2
2
.
here
so limit does not exist hence function is not continuous at 3
Rapid Questions Verification Of Continuity Of Functions: -
1.
Verify Continuity of function
f
(
x
) =
at
x
= 4.
2.
Verify Continuity of function
f
(
x
) = 3{
x
}+1 at
x
= 1. ({
x
} =
x
- [
x
])
Differentiability
Differentiability of a function at any point a is defined by
.
If above limit exists then function would be differentiable else not, above limit can be further written as
. In above if equality holds and each limit exists then function would be differentiable
Example based on differentiability of function using direct method
Example:
Check Differentiability of
f
(
x
) =
at
x
= 5
Sol.
For differentiability
. must exist which could be further written as
Hence function is differentiable at
x
= 5
Checking Differentiability by implying limit to zero using substitution method
Implication could be done either by
x
=
a
+
h
or
x
=
a
–
h
, let’s explore both of them
Substitution by
x
=
a
+
h
If substitution is done in the Right-hand limit.
In above limit
a
+
means moving towards a from right hand side
If substitution is done in the Left-hand limit.
In above limit a
–
means moving towards a from Left hand side
Substitution by x=a- h
If substitution is done in the Right-hand limit.
If substitution is done in the Left-hand limit.
Let
x
=
a
–
h
Example based on differentiability of functions using substitution method
Example: -
Check differentiability of
at
?
Sol.
sin
2
x
is always continuous function hence
is sufficient for checking differentiability
Let x=
Rapid Question based on differentiability of functions
1.
Check differentiability of
at
.
2.
Check differentiability of
at
.
Important Note :
if a function is differentiable then it must be continuous because it must hold
property, but if a function is continuous then it may be differentiable or not because
form doesn’t make sure the result to be finite.
