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The function is .
The function is continuous over the interval , and .
Check the continuity of the function at .
Definition of continuity :
A function is continuous at , if then it should satisfy three conditions :
(1) is defined.
(2) exists.
(3) .
Check the continuity of the function at .
Condition (1):
is defined.
If then the function is .
.
Condition (2):
exists.
Left hand limit :
Right hand limit :
Left hand limit and right hand limit are not equal, limit does not exist.
does not exist.
The limit does not exist at .
Therefore the function is discontinuous at .
and .
.
Therefore the function is continuous at from right side.
Check the continuity of the function at .
Condition (1):
is defined.
If then the function is .
.
Condition (2):
exists.
Left hand limit :
Right hand limit :
Left hand limit and right hand limit are not equal, limit does not exist.
does not exist.
The limit does not exist at .
Therefore the function is discontinuous at .
and .
.
Therefore the function is continuous at from left side.
Graph :
Graph the function :
The function is discontinuous at and it is continuous from right side.
The function is discontinuous at and it is continuous from left side.
Graph of the function is
.
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