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Mathematics : Continuity and Differentiability

Short Answer Type

91. Discuss the continuity of the function f given by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 plus straight x space comma space if space straight x not equal to 0 end cell row cell space space space space space 2 space space comma space if space straight x equals 0 end cell end table close at space straight x equals 0

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92. Discuss the continuity of the function

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell space space space space space space space straight x space space comma space 0 less or equal than straight x less than 1 half end cell row cell straight x over 2 space comma space straight x equals 1 half end cell row cell space 1 minus straight x space space comma space 1 half less than straight x less or equal than 1 end cell end table close at space straight x equals 1 half

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93. Discuss the continuity of the functions

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell space space space space space space straight x space space comma space space space straight x greater than 0 end cell row cell space space space space space 1 space space space comma space space space straight x equals 0 end cell row cell space minus straight x space space space comma space space space straight x less than 0 end cell end table close at space straight x equals 0

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94. Discuss the continuity of the function

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 1 half minus straight x space space comma space 0 less or equal than straight x less than 1 half end cell row cell 1 space space space space space space comma space straight x equals 1 half end cell row cell 3 over 2 minus straight x space space comma 1 half less than straight x less or equal than 1 end cell end table close at space straight x equals 1 half

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95.

Let
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 1 minus straight x space space space comma space space 0 less or equal than straight x less than 1 half end cell row cell 0 space space space comma space space straight x equals 1 half end cell row cell space space space straight x space space space space space space comma space space 1 half less than straight x less than 1 end cell end table close$
Discuss continuity of f(x) $at space straight x equals 1 half$

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96. Find the relationship between a and b so that the function f defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight a space straight x plus 1 comma space space space if space straight x less or equal than 3 end cell row cell straight b space straight x plus 3 comma space space space if space straight x greater than 3 end cell end table close

is continous at x=3

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97. For what value of λ is the function defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight lambda open parentheses straight x squared minus 2 straight x close parentheses comma space if space straight x less or equal than 0 end cell row cell 4 straight x plus 1 comma space space space space space space space if space straight x greater than 0 end cell end table close

continous at x=0? What about continuity at x=1?

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98.

In the following , find the value of the constant k so that the given function is continuous at the indicated point:
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight k space straight x plus 5 space space comma space if space straight x less or equal than 2 end cell row cell straight x minus 1 space space space space space comma space if space straight x greater than 2 end cell end table close at space straight x equals 2.$