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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) $at space straight x equals 1 half$

We space have space 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 space Lt with straight x rightwards arrow 1 half to the power of minus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 1 half to the power of minus below left parenthesis 1 minus straight x right parenthesis space space space space space space space space space space space space space space space open square brackets Put space straight x equals 1 half minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 1 half to the power of minus close square brackets space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below open curly brackets 1 minus open parentheses 1 half minus straight h close parentheses close curly brackets equals Lt with straight h rightwards arrow 0 below open parentheses 1 half plus straight h close parentheses equals 1 half plus 0 equals 1 half Also space straight f open parentheses 1 half close parentheses equals 0 therefore space Lt with straight x rightwards arrow 1 half to the power of minus below space straight f left parenthesis straight x right parenthesis not equal to straight f open parentheses 1 half close parentheses therefore space straight f left parenthesis straight x right parenthesis space is space discontinous space 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:
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight k space straight x plus 5 space space comma space if space straight x less or equal than 2 end cell row cell straight x minus 1 space space space space space comma space if space straight x greater than 2 end cell end table close at space straight x equals 2.$