Continuous Function : Original Concept : Real Analysis : B.sc.Mathematics #realanalysis #continuousfunction

For a video explanation, click here 👉 https://youtu.be/YKNZT-lU7_w

  Here in this page , i completely explained the original meaning of a continuous function at a point on the domain in detail. 

 CONTINUOUS FUNCTION :  

 Definition : 

A real valued function f : X → R is said to be continuous at a point a in X if " for a given

 𝞊 > 0 ∃ 𝝳 > 0 such that | f (x) - f(a) | < 𝞊 whenever | x-a | < 𝝳 .

 now here is the explanation of the above definition.

  Before going to the definition, we should learn two things.

   1.  | x - y | means the distance between x and y.

   2 .  | x-a | < r means x lies in circle with center a and radius r.

   Note : Here circle means the neighbourhood.

       For convinince or easily understand, here iam considering a neighbourhood in the real line  as a circle.   

                                                

Here a is in the domain X. Since there is a function f on X, f(a) lies in the codomain R. Now draw a circle M around f(a) with radius  𝞊 >0. Corresponding to this  𝞊, there exist 𝝳>0. 

Draw a circle N  around a with radius 𝝳 in the domian X. 

Now we call f is continuous at a in X if we take any x in N , its image f(x) must lie inside    M. 

 We see,  take any x in N i.e. | x-a | < 𝝳 , then its image f(x) must lies inside M 

 i.e. | f(x) - f(a) | < 𝞊. 

                                                      *** 

* finite vs small set

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