Continuity & Differentiability

Definitions and Formulas

Continuous Function:-  

(1) If f is a real function on a subset of real numbers and let c be a point in the domain of f. Then f is continuous at c if

Means, if the left-hand limit, right-hand limit, and the value of the function at x=c exists and also equal to each other, then f is said to be a continuous function at x=c.

OR

In general, if the value of the function at x=c equals the limit of the function at x=c then the function is called a continuous function.

(2) A real function f is said to be continuous if it is continuous at every point in the domain of f.

Discontinuous Function:- If the value of the function at x=c does not equal the limit of the function at x=c then the function is called a discontinuous function.

Algebra of continuous Function:- Suppose f and g be two real functions continuous a real number c. Then

(1) f+g is continuous at x=c.

Proof:- Given f and g be two continuous functions at x=c then

Hence, f+g is also a continuous function.

(2) f-g is continuous at x=c.

Hence, f-g is also a continuous function.

(3) f.g is continuous at x=c.

Hence, f.g is also a continuous function.

(4) f/g is continuous at x=c.  (Provided g(c) doesn’t equal to zero.)

Hence, f/g is also a continuous function.

NOTE:- If f and g are real valued functions such that (fog) is defined at c. If g is continuous at c and if f is continuous at g(c), then (fog) is also continuous at c.

Differentiability:- Let f is a real function and c is a point in its domain. The derivative of f at c is defined by

provided this limit exists. 

NOTE:- Derivative of f at c is denoted by f’(c) or d(f(x))/d(x). Then,

The process of finding derivates is known as differentiation and phrase differentiate f(x) with respect to x is known as f’(x).

Algebra of derivatives:- 

(1) (u±v)’=u’+v’

(2) (uv)’=u’v+uv’ (Leibnitz Rule or Product Rule)

(3) (u/v)’=(u’v-uv’)/v2 (provided v doesn’t equal to 0) (Quotient Rule)

NOTE:- If a function f is differentiable at point c, then it is also continuous at that point.

Proof:- Given f is differentiable at c, then

Derivatives Formulas:-

Derivatives of composite function(Chain Rule):- 

For Example:- f(x) = (2x+1)3

Derivatives of implicit function:- 

For Example:- Find dy/dx, if y+siny = cos x.

Solun:- Differentiate w.r.t x:-

Derivatives of inverse trigonometry function:-

Logarithmic Differentiation:-

Differentiate w.r.t. x:-

Then calculate dy/dx.

Derivatives of Functions in Parametric Form:- Let x=f(t) and y=g(t) is said to be a parametric form with t as a parameter

⇒ x=f(t)

Differentiate w.r.t. t:-

..........(1)

⇒ y=g(t)

Differentiate w.r.t. t:-

...........(2)

Eq. 2/Eq. 1:-

Second Order Derivative:- Second order derivative is denoted by f’’(x), d2y/dx2, D2y, y’’, y2.

For example:- Find d2y/dx2, if y=x3+tan x.

Solun:- Given y=x3+tan x

Derivatives of Logarithmic and Exponential Functions:-

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