Continuity and Differentiability





Definition of continuity at a point

 

A function f(x) is said to be continuous at x=a if given \epsilon > 0 , there exists \delta . 0 such that | f(x) – f(a) | < \epsilon \forall x such that | x –a | < \delta.

 

 

Alternative definition: f(x) is said to be continuous at x =a if value of f(x) = limit of f(x) at x = a

Or if lim f(x) = f(a)

x \rightarrow a

 

Of if lim f (a+h) = lim f(a –h) = f(a) …….Equation   1

If the condition (1) is not satisfied, then f(x) is said to be discontinuous at x =a.

Note: In order to rest continuity of a function at a point, we verify the equation (1). This is the working rule of continuity.

 

Definition of continuity in an interval

 

 

Let ‘h’ be a small positive number. A function f(x) is said to be continuous in closed interval [a,b], i.e., a \leq x \leq b if:

(i) f(x) is continuous \forall x in open interval (a ,b), i.e.,

 

Continuity and Differentiability

Continuity and Differentiability


a , x <b

(ii) f(x) is continuous at x = 1 from right

\underset{x \to a}{ [ lim} f ( a + h ) = f (a) ]

 

(iii) f(x) is continuous at x=b from left

\underset{x \to b}{ [ lim} f ( b - h ) = f ( b ) ]

 

Kinds of discontinuity

 

(1) Removable Discontinuity: A function f(x) is said to have removable discontinuity at x=a if \underset{x \to a}{lim} f(x) exists or \underset{h \to 0}{lim} f ( a + h ) = \underset{h \to 0}{lim} f ( a - h ) but \underset{x \to a}{lim} f ( x ) \neq f ( a ) . In this case value of function and limit of function are not equal.

 

(2) Discontinuity of First Kind: If f ( a - 0 ) = \underset{h \to 0}{lim} f ( a - h) \, \, \, and \, \, \, f ( a + 0 ) = \underset{h \to 0}{lim} f ( a + h) both exist and f (a -0) \neq f (a +0), then f(x) is said to have discontinuity of first kind or ordinary discontinuity at x=a.

 

(3) Discontinuity of second kind: A function f (x) is said to have discontinuity of second kind at x = a if f(a-0) or f (a + 0 ) or both do not exist.

 

 

Properties of continuous function

 

 

(1) if f is continuous on a closed interval [a,b], then it is bounded in this interval.

 

(2) if f is continuous in [a,b] and f(a) and f(b) have opposite signs, then there is at lest one value of x=c such that f(c) = 0 and a<c<b.

 

(3) If a function f is continuous in closed interval [a,b], then f(x) takes at least once all values of between f(a) and f(b).

 

Important note: If there is no gap in the graph of the function in a certain range [a,b], then f(x) is said to be continuous in [a,b].

 

 

Some standard continuous functions

 

(1) Every constant function f ( x ) = c \forall x is everywhere continuous.

 

(2) The identity function I (x) = x is everywhere continuous.

 

(3) The modulus function f(x) = |x| is continuous \forall x .

 

(4) The exponential function f(x) = a^x \forall x \epsilon R and a >0 is continuous everywhere.

 

(5) The logarithmic function f ( x ) = log _a x is continuous \forall x > o \, \, and \, \, a \neq 1 , a > 0

 

(6) Every polynomial function:  f ( x ) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

 

Differentiability

 

 

A function f(x) is said  to be differentiable at x = a if \underset{x \to 0}{lim} \dfrac{f ( x ) - f ( a )}{x - a} exists and is finite. Let ‘h’ be a small positive number.

 

Right hand derivative of f(x) at x = a is denoted by Rf’ (a) and is defined as Rf’ (a) =  \underset{h \to 0}{lim} \dfrac{f ( a + h ) - f ( a )}{h}

 

Left hand derivative of f(x) at x = a is denoted by Lf’ (a) and is defined as Lf’ (a)= \underset{h \to 0}{lim} \dfrac{f ( a - h) - f (a )}{ - h}

 

Alternate definition: A function is said to be differentiable at x = a if Rf’ (a) and Lf’ (a) both are finite and Rf’ (a) = Lf’ (a).

or \, \, \underset{h \to 0}{lim} \dfrac{f ( a + h ) - f ( a )}{h} = \underset{h \to 0}{lim} \dfrac{f ( a - H ) - f ( a )}{-h}

 

The common value is denoted by f’ (a).

 

 

Relation between continuity and differentiability

 

 

(1) if f(x) is differential at x = a, then it is continuous at x =a.

 

(2) if f(x) is continuous at x = a, then there is no guarantee that f (x) is differentiable at x=a.

 

(3) If f (x) is not differentiable at x =a, then it may or may not be continuous at x=a.

 

(4) if f(x) is not continuous at x=a, then it is not differentiable at x=a.

 

 

Some important results on differentiability

 

(1) Every polynomial function, constant function, exponential function is differentiable \forall x \epsilon R .

 

(2) if f(x) and g(x) are differentiable, then f (x) \pm g ( x ) , f ( x ) g ( x ) , \dfrac{f ( x )}{g ( x )} are differentiable functions provided in the last case g(x)\neq 0 .

 

(3) The composition of differentiable function is differentiable function.

 

Darboux Theorem

 

 

If f(x) is differentiable in the closed interval [a,b] anf f’ (a) and f’ (b) are of opposite signs, then there is a point c \epsilon ( a , b ) , ie., a < c < b such that f’ (c) = 0.

 

Graphical Meaning of Differentiability

 

 

F’(a) represents slope of tangent at x=a. A function f(x) is said to be differentiable at x=a if it (tangent) has unique slope at x = a.



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