THEOREM OF NECESSARY CONDITION DIFFERENTIABILITY OF A FUNCTION

THEOREM OF NECESSARY

CONDITION
 DIFFERENTIABILITY OF A FUNCTION

Theorem  :

Let f be a real valued function defined on a neighbourhood of
(a, b). If f differentiable at (a, b), then

(i) f is continuous at (a, b).
(ii) fx(a, b) and fy(a, b) both exist

Proof :

Let (a + h, b + k) be any point in the neighbourhood of (a, b). Since
f differentiable at (a, b).

∴ f(a + h, b + k) − f(a, b) = Ah + Bk + hφ(h, k) + kψ(h, k)
where A, B are constants independent of h, k and φ(h, k) → 0, ψ(h, k) → 0
as (h, k) → (0, 0)

(i) Taking limit as (h, k) → (0, 0) on both sides of the result     …. ……    (1).
∴ lim               [f(a + h, b + k) − f(a, b)]
    (h,k)→(0,0)    

=  lim         Ah + Bk + hφ(h, k) + kψ(h, k)       

    (h,k)→(0,0)

= 0
           lim            f(a + h, b + k) = f(a, b)
⇒ (h, k) → (0, 0)

⇒ f is continuous at (a, b).

(ii) Putting k = 0 in (1)

f(a + h, b) − f(a, b) = Ah + hφ(h, 0)

f(a + h, b) − f(a, b)
—---------------------     = A + φ(h, 0)

           h

  ∴ lim        f(a + h, b) − f(a, b)       
     h→0      —-----------------------
                              h
= 
      lim       
     h→0       (A + φ(h, 0)

=    A

   fx(a, b) = A

Similarly, we can show that
fy(a, b) = B
follows that fX(a, b) and fy(a, b) both exist.