Continuity And Differentiability


A function ‘f’ is continuous at x = p if it satisfy the following three conditions.

F (p) is defined;

Limxp f (p) exists;

Limxp f (p) = f (a);

If the function satisfies these three conditions then we can say that the function is continuous.

If the function is not continuous at x = p then we can say the function is discontinuous. In open interval (p, q) the function ‘f’ is continuous then the function is known as continuous on the interval (p, q).

If the addition of function ‘f’ is Limxp+ f (x) = f (p) and Limxq- f (x) = f (q) then the given function ‘f’ is continuous in the close interval [p, q]. If the function is polynomial then these polynomial function are continuous. Let us have a rational function    f (p).
                  g (q)

Where the function f (p) and f (q) are polynomial function is continuous everywhere except at p = r where the function g (r) = 0.

Let us take a function f (p) which is given by:

F (p) = u (p), if p < a,

            v (p), if p ≥ a,

Where u (p) is continuous for every p < a and v (p) is continuous for every p > a then function ‘f’ is continuous everywhere which provided ‘f’ is continuous at p = a.

If the function ‘f’ is differentiable at p0 then we can say that the function ‘f’ is continuous at p0. One condition for differentiable function, it must be continuous in its domain at every point. If we have a function which is continuous then we check Continuity and Differentiability of a function. It is not necessary that the continuous function be differentiable. The continuity and differentiability calculus includes the Functions, Sets and Relations while it is considered as a part of Algebra.

This is all about differentiability and continuity.

Math Topics