# Limit & Continuity

Hey there! In this post,we discuss about the concepts and definitions of the topic Limit & continuity. Starting from interval, function to the ε-δ definitions of the limit.

Starting from the basic definitions in this topic we’ll go step wise further.

#### Intervals:

There are three types of intervals.viz;

- Closed Interval
- Open interval
- Semi open or Semi closed intervals

Let a,b∈R and let, a<b,Now we define the types of intervals:

#### Closed interval:

[a , b] ={x| a ≤ x ≤ b } in which a & b are involved.

#### Open interval:

(a , b)={x| a < x < b } in which a & b are not involved.

#### Semi open interval:

We define semi open or semi closed intervals as;

[a ,b)={x| a ≤ x < b } or (a , b]={x| a < x ≤ b }

#### Absolute Value:

The absolute value is any real number x calculated as,|x| (mod x)

|x| = {_{-x/x<0}^{x/x0} =max{x ,-x}

#### Bounded set:

A non empty set is bounded if ∃ a real number M s.t.|a| ≤ M for every a ∈ A.

If a set is not bounded, then it is unbounded.

#### Meaning of |x-x_{0}| <δ :

x_{0} is a fixed real number and δ is a positive real number.

#### δ- neighborhood:

Let x_{0} be a fixed real number and δ be any positive real number. A set of numbers x, which differ from x_{0 }by an amount less than or greater than δ, is a δ neighborhood of x_{0}.

∴ N δ (x_{0}) = {x: |x-x_{0}|}

⇒x_{0} x_{0} N(x_{δ}) and N_{δ}=(x_{0}-δ, x_{0}+δ)

#### Deleted δ neighborhood:

When |x-x_{0}|>0, we have x ≠ xx_{0}. Thus the point x_{0} is deleted from N_{δ} (x_{0}).This is done by combining |x-x_{0}|>0 & |x-x_{0}|<δ

i.e.0<|x-x_{0}|<δ means x_{0}-δ <x< x_{0}; x_{0}<x<x_{0}+δ

#### Function:

Function is some kind of connecting rule between two non empty sets.

#### odd function:

If f is an odd function, then; f(-x) = -f(x)

#### Even function:

If f is even function then; f(-x) = f(x)

#### Real Valued function:

A real valued function has its range is a subset of the set of real numbers (R)

#### Constant Function:

If f(x) = constant, then the function f is constant function.

#### Zero Function:

If f(x) = 0; for every x∈X, then f is a zero function. A zero function is also a constant function.

#### Bounded Function:

A function f : A→R is bounded in A if there exist a constant M s.t. |f(x)| ≤ M for every x∈A.

#### Limit of a function:

we define limit of a function as:

lim_{x→x0} f(x) = l.

#### ε-δ definition of limit:

For any given positive number ε, there exist a positive number δ s.t.

0 < | x-x_{0} | < δ, x ∈ [a,b] ⇒| f(x)-l | < ∈

In above context, the essential & basic concepts of limit are discussed. Now we move towards the axioms and some important theorems in the limit.