Continuity And Differentiability In Mathematics | Mathquery

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     In mathematics continuity and derivability or differentiability plays an important role . So let us discuss it .

Definition Of Continuity Of A Function :-

       Let f be a function defined on an interval [a,b] . We shall now consider the behaviour of f at points of [a,b] .

Continuity At A Point :

 Definition(Continuity At An Internal Point) :

         A function f is said to be continuous at a point c , a<c<b , if 

                    lim f(x) = f(c)

                   x-->c

   In other words , the function is continuous at c , if for each ε>0 , there exists δ>0 such that 

             |f(x) - f(c) | < ε , when |x-c|<δ

Here lim f(x) is called the limit of a function

         x-->c 

i.e. the function exists when x tends to c.

(i)  A function f is said to be continuous from   the left at c if 

               lim f(x) = f(c) 

            x-->c-0

(ii)  Also f is continuity from the right at c if 

                lim  f(x) = f(c)

              x-->c+0

   Clearly a function is continuous at c iff it is continuous from the left as well as from the right .

Definition(Continuity At An End Point):

                   A function f is defined on a closed interval [a,b] is said to be continuous at the end point a if it is continuous from the right at a , i.e.

                  lim f(x) = f(a)

                 x-->a+0

  Also the function is continuous at the end point b of [a,b] if 

               lim f(x) = f(b)

            x-->b-0

   Thus a function f is continuous at a point c if 

(i)  lim f(x) exists  , and 

     x-->c

(ii)  limit equals the value of the function at         x= c.

        Let's discuss about discontinuity of a function .

Example :

      Show that the function defined by 

                               { x sin 1/x , when x≠0

                     f(x) ={

                               {    0 ,   when x= 0

is continuous at x = 0.

Solution :

          Now 

                        lim f(x) = lim (x sin 1/x) = 0

                      x-->0

so that 

                       lim f(x) = f(0) 

                     x-->0

   Hence f is continuous at x = 0 .  (Proved)

Discontinuous Function :

                A function is said to be discontinuous at a point c of its domain if it is not continuous there at c. The point c is then called a point of discontinuity of the function .

                   Now let us discuss about Uniform continuous  of a function in mathematics .

Definition Of Uniform Continuity:

       A function f defined on an interval I is said to be uniformly continuous on I if to each ε>0 there exists a δ>0 such that 

   |f(x₂) - f(x₁)| < ε , for arbitrary points x₁,x₂ of I for which |x₁-x₂|<δ.

Example :

    Prove that f(x) = sin x² is not uniformly continuous on [0,∞[ .

Solution :

      Let ε = 1/2 and δ be any positive number such that for n>π/δ² 

                           |√(nπ/2 ) - √(n+1)π/2| <δ

    Therefore , taking x₁ = √(nπ/2 ) and         x₂= √(n+1 )π/2 , as any two points of the interval [0,∞[.

|f(x₁) - f(x₂)| = | sin nπ/2 - sin (n+1)π/2 | = 1

                                                                         >ε,

          |x₁-x₂| <δ 

Hence f(x) = sin x² is not uniformly continuous on [0,∞[ .

 Differentiability At A Point :

            Let f be a real valued function defined on an interval I = [a,b] ⊂ R . It is said to be derivable at an interior point c (where a<c<b) if 

          lim  f(c+h) - f(c) / h 

          h-->0

 or     lim f(x) - f(c) / x-c  exists .

          x-->c

     The limit in case it exists , is called the Derivative or the Differential Coefficient of the function f at x = c , and is denoted by f'(c) . The limit exists when the left - hand and the right - hand limits exists and are equal .

     lim f(x) - f(c) / x-c 

   x-->c-0

      is called the left - hand Derivative and is denoted by 

           f'(c-0) , f'(c⁻) or Lf'(c) 

While  lim f(x) - f(c) / x-c 

            x-->c+0   

        is called the right - hand Derivative and is denoted by 

         f'(c+0) , f'(c+) or R f'(c) 

Thus , the derivative f'(c) exists when 

            Lf'(c) = Rf'(c) 

Example : 

         Show that the function f(x) = x² is derivable   on [0,1] .

Solution : 

            Let x₀ be any point of ]0,1[, then 

   f'(x₀) = lim x²-x₀² / x-x₀ = lim(x+x₀) = 2x₀

              x-->x₀                     x-->x₀

  At the end points , we have 

 f'(1) = lim f(x)-f(0) / x-0 = lim x²/x = lim x =0

          x-->0+0                      x-->0+

f'(1) = lim f(x)-f(1) / x-1 = lim x²-1 / x-1 

          x-->1-0                     x-->1⁻

         = lim (x+1) =2

           x-->1⁻

Thus , the function is derivable in the closed interval [0,1].

References :-

(I).

         By Abhijit Debnath.

(II).

       (Dover Books on Mathematics)