Continuity & Differentiability

Continuity & Differentiability

Some Important Results on Limits

If p(x) is a polynomial, p(x) = p(a).

, a > 0

.


= .


cosx = 1 (where 'x' is in radians)





For two function within the same domain, we say that f < g if f(x) < g(x) for all x in the domain.


If both exist, then f g .


If and f, g, h are real functions such that f g h for all points in an open interval containing a, then . (Sandwitch

theorem)


If for all points in an open interval containing a, 0 f g and , then .


if and only if .

Remarks:

(i)Often, f(x) can be simplified by using series expansion, rationalization or using conjugate surds, before limit is obtained.

(ii)For the existence of the limit at x = a, f(x) need not be defined at x = a. However if f(a) exists, limit need not exist or even if it exists then it need not be equal to

f(a).

Frequently Used Series Expansions

Following are some of the frequently used series expansions:

sin x =

cos x =

tanx =

e^x =

a^x = 1 + x.lna + (lna)² + .... , a R⁺

(1+ x)ⁿ = 1 + nx + n R. |x|<1, n is any real number

ln(1+ x) = - 1 < x 1.

= e








= ln a, a R.

= = = log_ae; a > 0, a 1.

= 1

.

.

.

Note: If then the following results hold true:

cos f(x) = 1


= 1


= ln b ( b> 0)


= e

Limit at infinity

In case we want to find the limit of a function f (x) as x takes larger and larger values, we write ..

If the degree of the denominator in a rational (polynomial) function is higher than the degree of the numerator, then the is zero. If the degree of the

denominator is lower than the degree of the numerator, then the .




if .

Algebra of limits

The following are some of the Basic Theorems on limits which are widely used to calculate the limit of the given functions.

Let f(x) = ₁ and g(x) = ₂ where ₁ and ₂ are finite numbers. Then

(c₁f(x) c₂g(x)) = c₁₁ c₂₂, where c₁and c₂ are given constants.


f(x).g(x) =f(x) g(x) = l₁. l₂.


.


f(g(x)) = f(g(x)) = f(₂), if and only if f(x) is continuous at x = ₂.


In particular, ln(g(x)) = ln(g(x)) = ln ₂ if ₂ > 0.



Note: If exists, then it is not always true that will exist.

 

L' Hospital's Rule

We state below a rule, called L' Hospital's Rule, meant for problems on limit of the indeterminante form 0/0 or /. The other indeterminate forms are - , 0.

, 0⁰, ⁰, 1.


Continuity

A function f(x) is said to be continuous at x = a if = f(a) i.e. L.H.L.= R.H.L. f(a) = value of the function at a i.e..

If f(x) is not continuous at x = a, we say that f(x) is discontinuous at x = a.

f(x) will be discontinuous at x = a in any of the following cases:

exist but are not equal.


exist and are equal but not equal to f(a).


f(a) is not defined.


At least one of the limits does not exist.


Properties of Continuous Functions

Let f(x) and g(x) be functions, both continuous at x = a. Then

cf(x) is continuous at x = a where c is any constant.

f(x) g(x) is continuous at x = a.

f(x) . g(x) is continuous at x = a.

f(x) / g(x) is continuous at x = a, provided g(a) 0.

These results also hold in any interval, if both f (x) and g (x) are continuous in that interval.


Continuity in an Interval

f(x) is said to be continuous in an open interval (a, b) if it is continuous at every point in this interval.

f(x) is said to be continuous in the closed interval [a, b] if

f(x) is continuous in (a, b)








If f (x) is continuous, then a small change in x produces a small change in f (x).


Let f (x) be continuous in the interval (a, b). Then, while passing from one value x₀ of x to another x₁ in (a, b), the function f (x) must take every

intermediate value in (f (x₀), f (x₁)), atleast once.

If f (x) is continuous in (a, b), then the graph of y = f (x) cannot have any breaks in (a, b).

Function f(x)	Interval in which f(x) is continuous.
Constant C xn, n is an integer 0. |x - a| x-n, n is a positive integer. a0xn + a1xn-1+...+an-1x + an p(x)/q(x), p(x) and q(x) are polynomials in x sin x cos x	(- , ) (- , ) (- , ) (- ,) - {0} (- , ) R - {x :q(x) = 0} R R
tan x	R -
cot x	R -
Secx	R -
cosecx	R -
ex	R
ln x	(0, )

Continuity of Composite Functions

If the function u = f(x) is continuous at x = a, and the function y = g(u) is continuous at

u = f(a), then the composite function y = (gof)(x) = g(f(x)) is continuous at x = a.

Removable discontinuity

If exists but is not equal to f(a), then f(x) has removable discontinuity at x = a and it can be removed by redefining f(x) for x = a.


Non-removable Discontinuity

If f(x) does not exist, then we cannot remove this discontinuity so that this becomes a non- removable or essential discontinuity


Differentiability

Let y = f(x) be a continuous function of x in (a, b). Then the derivative or differential coefficient of f(x) w.r.t. x at x (a, b), denoted by dy/dx or f '(x), is

... (1)

provided the limit exists and is finite; and the function is said to be differentiable.


To find the derivative of f(x) from the first Principle

If we obtain the derivative of y = f(x) using the formula , we say that we are finding the derivative of f(x) with respect to x from

the definition or from the first principle.


Right Hand Derivative

Right hand derivative of f(x) at x = a is denoted by, Rf'(a) or f'(a⁺) and is defined as

R= , h > 0.


Left Hand Derivative

Left hand derivative of f(x) at x = a is denoted by or and is defined as

= , h > 0.

Clearly, f(x) is differentiable at x = a if and only if R f '(a) = Lf '(a).

If the function y = f (x) is differentiable at x = x₀, it must be continuous at x = x₀.


A function which is continuous at x = x₀ may or may not be differentiable at x = x₀.

Rolle' Theorem:


If a function f (x) is

(i)continuous in a closed interval [a , b],

(ii) differentiable in the open

interval (a , b),

(iii) f (a) = f (b),then there exist atleast one value c (a, b) such that f' (c) = 0.





Note:
If we write y = f (x) and represent (a, f (a)) and (b, f (b)) by points A and B on the curve

y = f (x), the ordinates of the points A and B are equal. The Rolle' Theorem concludes that there is a point on the curve, the tangent at which is parallel to

the chord AB.


The conclusion of Rolle' Theorem may not hold good for a function which does not satisfy any of its conditions. Consider the function y = f (x) = |x| in the

interval [-1, 1]. Here the function is continuous in [-1, 1] and f (1) = f (-1). But f (x) is not differentiable at x = 0 the function is not differentiable in (-1,

1). Hence Rolle' Theorem can't be applied in this case.



Lagrange' Mean Value Theorem:

If a function f (x) is

(i) continuous in a closed interval [a , b] and

(ii) differentiable in the open interval (a , b),

then there is at least one value of x = c (a, b) such that. .... (1)

Note:

If we write b - a = h so that h denotes the length of the interval [a, b], we can write

c = a+ h, where (0, 1). Thus equation (1) becomes f (a + h) = f (a) + h f' (a + h) for 0 << 1.


Let A (a, f (a)), B (b, f (b)) and C (c, f (c)) be points on the curve y = f (x) such that.
The slope of the chord AB = and that of the tangent at C is f' (c). These being equal, it follows that there exists a point c on the curve y = f (x), the

tangent at which is parallel to the chord AB.

is the average rate of change of f (x) over the interval [a, b] and f' (c) is the actual rate of change of f (x) at x = c. Thus the theorem states that

the average rate of change of f (x) over [a, b] is actually achieved at least for one value of x (a, b).