Continuity And Differentiability

Definition of Continuity

Continuity defined as continuous transfer of information from one function to another function.

• A continuous variable is nothing but within the limit the variable changes, any value is possible. Non continuous variables are called discrete variables. • continuous variable can be obtaining any value on the scale used to calculate it. • For any continuous random variable with the probability density functions f(x), the ∫ f(x) dx = 1 , for all the values of x. Example: f(x) = 3x for 0 x 1. Find the expected value continuous of given f(x). Solution: E (X) = X’ = x f(x) dx = x (3x) dx = 3x2 dx = [3/ 3 x3] = 1 - 0 E(X) = 1 Definition of Differentiability It is defined by the equivalent to finding a slope of the tangent line to the function at a point. The point P, is to find the slope of the tangent line to a graph. A process of determining a rate and change of a curve in the mathematical process of finding the derivative of a function and its derivative is xn is nxn - 1 Example: solve the x2-5x+1 Solution: The given differentiation is x2-5x+1 Differentiate with respect to ‘x’ d/dx=( x2-5x+1) =2x-5 Answer: 2x+5