Is the function defined by a continuous function?
The given function is,
f ( x ) = { x + 5 , x ≤ 1 x − 5 , x > 1
Consider k be any real number, then the cases will be k < 1 , k = 1 or k > 1 .
When k < 1 , the function becomes,
f ( k ) = k + 5
The limit of the function is,
lim x → k f ( x ) = lim x → k ( x + 5 ) = k + 5
It can be observed that, lim x → k f ( x ) = f ( k ) .
Therefore, the function is continuous for all real numbers less than 1.
When k = 1 , the function becomes,
f ( 1 ) = 1 + 5 = 6
The left hand limit of the function is,
LHL = lim x → 1 − f ( x ) = lim x → 1 − ( x + 5 ) = 6
The right hand limit of the function is,
RHL = lim x → 1 + f ( x ) = lim x → 1 + ( x − 5 ) = − 4
It can be observed that, LHL ≠ RHL .
Therefore, the function is discontinuous at x = 1 .
When k > 1 , then the function becomes,
f ( k ) = k − 5
The limit of the function is,
lim x → k f ( x ) = lim x → k ( x − 5 ) = k − 5
It can be observed that, lim x → k f ( x ) = f ( k ) .
Therefore, the function is continuous for all real numbers greater than 1.
Thus, only point of discontinuity is 1.
The given function is,
Consider
When
The limit of the function is,
It can be observed that,
Therefore, the function is continuous for all real numbers less than 1.
When
The left hand limit of the function is,
The right hand limit of the function is,
It can be observed that,
Therefore, the function is discontinuous at
When
The limit of the function is,
It can be observed that,
Therefore, the function is continuous for all real numbers greater than 1.
Thus, only point of discontinuity is 1.
