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<0x/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-x0| <δ :

x0 is a fixed real number and δ is a positive real number.

δ- neighborhood:

Let x0 be a fixed real number and δ be any positive real number. A set of numbers x, which differ from xby an amount less than or greater than δ, is a δ neighborhood of x0.
∴ N δ (x0) = {x: |x-x0|}
⇒x0 x0 N(xδ) and Nδ=(x0-δ, x0+δ)

Deleted δ neighborhood:

When |x-x0|>0, we have x ≠ xx0. Thus the point x0 is deleted from Nδ (x0).This is done by combining |x-x0|>0 & |x-x0|<δ

i.e.0<|x-x0|<δ means x0-δ <x< x0; x0<x<x0

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:
limx→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-x0 | < 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.

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