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A function f (x) is said to be continuous at x = a, if and only if:




  • The sum, difference and product of two continuous functions is a continuous function.
    This property holds good for any finite number of functions.
  • The quotient of two continuous functions is a continuous function provided the denominator is not equal to zero.

Discontinuous function

A function f (x) is said to be discontinuous at x = a for any of the following reasons:
  •  exists but it is not equal to f(a)
  • Description: 89531.png does not exist
  • f is not defined at x = ai.e.f(a) does not exist
  • If a function f (x) is not continuous at x = a, it is known as a discontinuous function.

The value of f(0) so that f(x= (1 5x)1/x is continuous at x = 0 is
Description: 89579.png
Description: 89594.png
Description: 89631.png 
But f(2) = k {f(x) = k when x = 2}
As f(x) is continuous at x = 2,
Description: 89647.png 


Examine the function  for continuity at x = 2.
HereDescription: 89722.png
Description: 89759.png 
So the right hand and left hand limits of f(x) as x → 2 are the same.
Description: 89771.png 
But the value of f(x) at x = 2 is 6 i.e.f(2) = 6
Description: 89781.png 
∴ The function is not continuous at x = 2

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