Continuity‌ ‌and‌ ‌Differentiability‌

Continuity

Continuity of a function is defined as the consistency of a function in a given interval. A function is said to be continuous if at no point, the function breaks or varies from other functional values of the given function. The continuity of a function is defined at every point of the function having the same value. This means that for a real valued function to be continuous, the function at every point in its domain should be continuous.

Continuity of a function f(x) at a point ‘a’ is expressed as

lim_x→a f(x) = f(a)

For example, consider the following function

f(x) = (x + 1)/2 if x is odd

and

f(x) = x/2 if x is even

The function f(x) will be continuous only if it is continuous for both odd and even values of x.

Conditions for continuity of a function

The following conditions are satisfied if a function is continuous:

1. A function which is defined for an open interval (a,b) will be continuous if at any point, say a, the function is continuous in the interval (a,b).
2. A function which is defined for a closed interval [a,b] will be continuous if at any point the function is continuous in the intervals (a,b) and [a,b].
3. If a function f(x) is continuous at a point a then its left hand limit (LHL) and right hand limit (RHL) will be equal to each other and the function at the point a.

   lim_x→a- f(x) = lim _x→a f(x) = lim _x→a+ f(x)

4. If there be two continuous functions f(x) and g(x) then
   1. f(x) + g(x) is also continuous
   2. f(x) – g(x) is also continuous
   3. f(x) . g(x) is also continuous
   4. f(x)/g(x) is also continuous for g(x) not equal to 0 at every point.

Differentiability

Differentiability of a function at any point is defined there exists the derivative of the given function at that point. Mathematically, differentiability for a function f(x) is

Lim_x→0 f’(x) = [f(x + h) – f(x)]/h

Where f’(x) is the derivative of f(x). It can also be denoted as d/dx . f(x). The method of obtaining the derivative of a function is called differentiation with respect to a variable. Here, the function f(x) is differentiated with respect to ‘x’.

Rules of differentiation

1. For a function xⁿ

   f(x) = xⁿ

   f’(x) = n . x^n-1

   is the required derivative.
2. Derivatives of standard trigonometric functions are
   1. d/dx . sinx = cosx
   2. d/dx . cosx = - sinx
   3. d/dx . tanx = sec²x
   4. d/dx . cotx = -cosec²x
   5. d/dx . secx = secx . tanx
   6. d/dx . cosecx = - (cotx . cosecx)
3. For functions u and v

   (u + v)’ = u’ + v’

4. For functions u and v

   (u - v)’ = u’ - v’

5. For functions u and v

   (u . v)’ = u’ . v + u . v’

6. For functions u and v

   (u/v)’ = [(u’ . v) – (u . v’)]/v²

   Where v is a not equal to 0.