Question

please answer this question..​

Answer 1

Given :   f(x)  =  x   if  x ≥ 0   ,           x²   if  x  <  0

To find : Check continuity

Solution:

f(x)  =  x   if  x ≥ 0

         x²   if  x  <  0

at x = 0

LHL = Lim h→0  f(0- h)  

= Lim h→0 (0 - h) ²  

= Lim h→0  h²

=  0

f(0) =  0

RHL = Lim h→0  f(0+ h)  

= Lim h→0  0 + h

= 0

LHL = f(0) = RHL

function is continues at x = 0

f(x)  =  x   for x > 0

Lim h→0  f(c+h)  = Lim h→0  f(c-h)

=> Lim h→0 c + h =  Lim h→0 c - h

=> c = c

function is continuous  for x  > 0

for x <  0

Lim h→0  f(c+h)  = Lim h→0  f(c-h)

Lim h→0   (c+h)²  = Lim h→0  (c-h)²

=> c² = c ²

function is continuous  for x  < 0

Hence function is continuous

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