Flashcards: Continuity and Differentiability | Mathematics (Maths) for JEE Main & Advanced PDF Download

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CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
A function ?? ( ?? ) is said to be continuous at a point ?? = ?? in the domain of ?? ( ?? ) if and only if ?? ( ?? -
) =
?? ( ?? +
) = ?? ( ?? ) = finite number, i.e., l im
?? ? ?? ? ?? ( ?? ) exists finitely, ?? ( ?? ) is a finite number and l im
?? ? ?? ? ?? ( ?? ) =
?? ( ?? ). More precisely, for given ?? > 0 , ?? > 0 such that 0 = | ?? - ?? | < ?? ? | ?? ( ?? ) - ?? ( ?? ) | < ?? 
  
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CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
A function ?? ( ?? ) is said to be continuous at a point ?? = ?? in the domain of ?? ( ?? ) if and only if ?? ( ?? -
) =
?? ( ?? +
) = ?? ( ?? ) = finite number, i.e., l im
?? ? ?? ? ?? ( ?? ) exists finitely, ?? ( ?? ) is a finite number and l im
?? ? ?? ? ?? ( ?? ) =
?? ( ?? ). More precisely, for given ?? > 0 , ?? > 0 such that 0 = | ?? - ?? | < ?? ? | ?? ( ?? ) - ?? ( ?? ) | < ?? 
  
 
 
 
CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
 
  
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CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
A function ?? ( ?? ) is said to be continuous at a point ?? = ?? in the domain of ?? ( ?? ) if and only if ?? ( ?? -
) =
?? ( ?? +
) = ?? ( ?? ) = finite number, i.e., l im
?? ? ?? ? ?? ( ?? ) exists finitely, ?? ( ?? ) is a finite number and l im
?? ? ?? ? ?? ( ?? ) =
?? ( ?? ). More precisely, for given ?? > 0 , ?? > 0 such that 0 = | ?? - ?? | < ?? ? | ?? ( ?? ) - ?? ( ?? ) | < ?? 
  
 
 
 
CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
 
  
 
 
 
 
Continuity of a Function in Open Interval 
A function is said to be continuous in an open interval ( ?? , ?? ), if it is continuous at each point of ( ?? , ?? ). 
  
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CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
A function ?? ( ?? ) is said to be continuous at a point ?? = ?? in the domain of ?? ( ?? ) if and only if ?? ( ?? -
) =
?? ( ?? +
) = ?? ( ?? ) = finite number, i.e., l im
?? ? ?? ? ?? ( ?? ) exists finitely, ?? ( ?? ) is a finite number and l im
?? ? ?? ? ?? ( ?? ) =
?? ( ?? ). More precisely, for given ?? > 0 , ?? > 0 such that 0 = | ?? - ?? | < ?? ? | ?? ( ?? ) - ?? ( ?? ) | < ?? 
  
 
 
 
CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
 
  
 
 
 
 
Continuity of a Function in Open Interval 
A function is said to be continuous in an open interval ( ?? , ?? ), if it is continuous at each point of ( ?? , ?? ). 
  
 
 
 
Continuity in Closed Interval 
A function ?? ( ?? ) is said to be continuous on a closed interval [ ?? , ?? ] if 
1 ?? ( ?? ) s continuous from right at ?? = ?? , i.e., 
l im
h ? 0
? ?? ( ?? + h ) = ?? ( ?? ) 
2 ?? ( ?? ) is continuous from left at ?? = ?? , i.e., 
l im
h ? 0
? ?? ( ?? - h ) = ?? ( ?? ) 
3 ?? ( ?? ) is continuous at each point of the open interval ( ?? , ?? ) 
  
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CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
A function ?? ( ?? ) is said to be continuous at a point ?? = ?? in the domain of ?? ( ?? ) if and only if ?? ( ?? -
) =
?? ( ?? +
) = ?? ( ?? ) = finite number, i.e., l im
?? ? ?? ? ?? ( ?? ) exists finitely, ?? ( ?? ) is a finite number and l im
?? ? ?? ? ?? ( ?? ) =
?? ( ?? ). More precisely, for given ?? > 0 , ?? > 0 such that 0 = | ?? - ?? | < ?? ? | ?? ( ?? ) - ?? ( ?? ) | < ?? 
  
 
 
 
CONTINUITY & DIFFERENTIABILITY 
Continuity of a Function at a Point 
 
  
 
 
 
 
Continuity of a Function in Open Interval 
A function is said to be continuous in an open interval ( ?? , ?? ), if it is continuous at each point of ( ?? , ?? ). 
  
 
 
 
Continuity in Closed Interval 
A function ?? ( ?? ) is said to be continuous on a closed interval [ ?? , ?? ] if 
1 ?? ( ?? ) s continuous from right at ?? = ?? , i.e., 
l im
h ? 0
? ?? ( ?? + h ) = ?? ( ?? ) 
2 ?? ( ?? ) is continuous from left at ?? = ?? , i.e., 
l im
h ? 0
? ?? ( ?? - h ) = ?? ( ?? ) 
3 ?? ( ?? ) is continuous at each point of the open interval ( ?? , ?? ) 
  
 
 
 
 
 
Properties of Continuous Function 
1 If ?? and ?? are continuous at ?? = ?? , then 
(a) ?? + ?? is continuous at ?? = ?? 
(b) ?? - ?? is continuous at ?? = ?? 
(c) ???? is continuous at ?? = ?? 
(d) ?? / ?? is continuous at ?? = ?? , provided ?? ( ?? ) ? 0 
(e) ???? is continuous at ?? = ?? , where ?? is any real constant 
(f) [ ?? ( ?? ) ]
?? / ?? is continuous at ?? = ?? , provided [ ?? ( ?? ) ]
?? / ?? is defined on an interval containing ?? , and 
?? , ?? are integers