continuity and differentiation

CONTINUITY and DIFFERENTIATION

CONTINUITY

1. A real valued function function f(x) is continuous at a point ‘a’

in its domain iff lim f (x) = f (a)

X a

i.e. the limit of the function at x = a is equal to the value of the

function at x = a.

2. A function f (x) is said to be continuous if it is continuous at

every point of its domain.

3. Sum, difference, product and quotient of continuous functions

are continuous i.e, if f (x) and g (x) are continuous functions on

their common domain, then f ± g, fg, f/g, kf (k is a constant)

are continuous.

4. Let f and g be real functions such that fog is defined. If g is

continuous at x = a and f is continuous at g (a), then fog is

continuous at x = a.

5. Following functions are everywhere continuous.

i) A constant function

ii) The identity function

iii) A polynomial function

iv) Modulus function

v) Exponential function

vi) Sine and Cosine functions

6. Following functions are continuous in their domains.

i) A logarithmic function

ii) A rational function

iii) Tangent, cotangent, secant and cosecant functions.

7. If f is continuous function, then |f| and 1/f are continuous in

their domains.

8. Sin-1x, cos-1 x, tan-1 x, cot-1 x, cosec-1 x and sec-1 x are

continuous functions on their respective domains.

DIFFERENTIATION

1. A real valued function f (x) defined on (a, b) is said to be

differentiable at x = c ϵ (a, b), iff

Lim f (x) – f (c ) exists finitely. x c x – c = Lim f (x) – f (c ) = Lim f (x) – f (c ) x c- x – c h c+ x – c = Lim f (c - h) – f (c ) = Lim f (c + h) – f (c ) = h 0 -h h 0 h

(LHD at x = c) = (RHD at x = c)

2. A function is said to be differential, if it is differentiable at every point in its domain.

3. Every differentiable function is continuous but, the converse is not necessarily true.

4. Following are some results on differentiability: i) Every polynomial function is differentiable at each x ϵ R. ii) The exponential function ax, a > 0, a ≠ q is differentiable at

each x ϵ R. iii) Every constant function is differentiable at each x ϵ R. iv) The logarithmic function is differentiable at each point in its

domain. v) Trigonometric and inverse-trigonometric functions are

differentiable in their respective domains. vi) The sum, difference, product and quotient of two

differentiable functions is differentiable. vii) The composition of differentiable function is a differentiable

function. viii) If a function f (x) is differentiable at every point in its

domain, then = Lim f (x + h) – f (x ) = Lim f (x - h) – f (x ) h 0 h h 0 -h Is called the derivative or differentiation of at x and is denoted by f’ (x) or d/dx (f(x)).

TYPE I ON TESTING THE CONYINUITY OF A FUNCTION IN ITS DOMAIN EXAMPLE 1 If a function f is defined as f(x) = |x-4|, x ≠4 x-4 Show that f is everywhere continuous except at x = 4 SOLUTION We have, f (x) = |x-4|, x ≠4 x-4 0, x = 4 -(x-4) = -1 ; x < 4 x-4 -(x-4), x < 4 |x-4| = x -4, x ≥ 4 x-4 = 1 x > 4

f(x) = x-4 0 ; x = 4 When x < 4 we have f(x) = -1, which, being a constant function, is continuous at each point x < 4. Also, when x > 4, we have f(x) = 1, which, being a constant function, is continuous at each point x > 4. Let us consider the point x = 4