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The given function is,
f ( x ) = { | x | + 3 , x ≤ − 3 − 2 x , − 3 < x < 3 6 x + 2 , x ≥ 3
Consider k be any real number, then the cases will be k < − 3 , k = − 3 , − 3 < k < 3 , k > 3 or k = 3 .
When k < − 3 , the function becomes,
f ( k ) = − k + 3
The limit of the function is,
lim x → k f ( x ) = lim x → k ( − x + 3 ) = − k + 3
It can be observed that, lim x → k f ( x ) = f ( k ) .
Therefore, the function is continuous for all real numbers less than − 3 .
When k = − 3 , the left hand limit of the function is,
LHL = lim x → 3 − f ( x ) = lim x → 3 − ( − x + 3 ) = − ( − 3 ) + 3 = 6
The right hand limit of the function at k = − 3 is,
RHL = lim x → 3 + f ( x ) = lim x → 3 + ( − 2 x ) = ( − 2 ) ( − 3 ) = 6
It can be observed that, LHL = RHL .
Therefore, the function is continuous at x = − 3 .
When − 3 < k < 3 , the function becomes,
f ( k ) = − 2 k
The limit of the function is,
lim x → k f ( x ) = lim x → k ( − 2 x ) = − 2 k
It can be observed that, lim x → k f ( x ) = f ( k ) .
Therefore, the function is continuous in − 3 < k < 3 .
When k = 3 , the left hand limit of the function is,
LHL = lim x → 3 − f ( x ) = lim x → 3 − ( − 2 x ) = − 6
The right hand side of the function at k = 3 is,
RHL = lim x → 3 + ( 6 x + 2 ) = 6 ( 3 ) + 2 = 20
It can be observed that, LHL ≠ RHL .
Therefore, the function is discontinuous at x = 3 .
When k > 3 , the function becomes,
f ( k ) = 6 k + 2
The limit of the function is,
lim x → k f ( x ) = lim x → k ( 6 x + 2 ) = 6 k + 2
It can be observed that lim x → k f ( x ) = f ( k ) .
Therefore, the function is continuous for all numbers greater than 3 .
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
When
The right hand limit of the function at
It can be observed that,
Therefore, the function is continuous at
When
The limit of the function is,
It can be observed that,
Therefore, the function is continuous in
When
The right hand side of the function at
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 numbers greater than
