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

$a , x

(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.

Related posts:

1. Continuity Theorems. Basic theorems of continuity or basic properties of continuity are...
2. Continuity of a function(continuous and discontinuous functions). Continuity of a function , a function can either be...
3. Partial Differentiation Defination of Partial Differentiation   If f is a function...
4. Second and higher derivatives. Think of a function y=f(x) , and let y=f(x) be...
5. Antiderivatives ( Indefinite Integrals) What is Antiderivative or Indefinite Integral? How is it calculated?...