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Continuity and Differentiability

Short Answer Type

81. If f is continuous and g is a discontinuous function then f + g is discontinuous function.

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Long Answer Type

82. Examine the continuity of f , where f is defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell sin space straight x minus cos space straight x comma space space space if space straight x not equal to 0 end cell row cell space space space space space space minus 1 space space space space space space space space space comma space space space if space straight x equals 0 end cell end table close

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83. Find all points of discontinuity of f where

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space straight x over denominator straight x end fraction comma space space if space straight x less than 0 end cell row cell space space straight x plus 1 comma space space space if space straight x greater or equal than 0 end cell end table close

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Short Answer Type

84. Let f be defined by

$straight f left parenthesis straight x right parenthesis open curly brackets table attributes columnalign left end attributes row cell space space space space space straight x squared over straight x space space space space space space space space space space space space comma space 0 space less or equal than straight x less or equal than 1 end cell row cell 2 straight x squared minus 3 straight x plus 3 over 2 comma space 1 less than straight x less or equal than 2 end cell end table close$
Show that f is continous at x=1.

Here space straight f left parenthesis straight x right parenthesis open curly brackets table attributes columnalign left end attributes row cell space space space space space straight x squared over straight x space space space space space space space space space space space space comma space 0 space less or equal than straight x less or equal than 1 end cell row cell 2 straight x squared minus 3 straight x plus 3 over 2 comma space 1 less than straight x less or equal than 2 end cell end table close space Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 1 to the power of plus below open parentheses straight x squared over 2 close parentheses space space space space space space space space left square bracket Put space straight x equals 1 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 to the power of minus right square bracket 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 fraction numerator left parenthesis 1 minus straight h right parenthesis squared over denominator 2 end fraction equals space Lt with straight h rightwards arrow 0 below fraction numerator straight h squared minus 2 straight h plus 1 over denominator 2 end fraction equals fraction numerator 0 minus 0 plus 1 over denominator 2 end fraction equals 1 half space Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below open parentheses 2 straight x squared minus 3 straight x plus 3 over 2 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket space Put space straight x equals 1 plus straight h comma space straight h greater than 0 right square bracket 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 2 left parenthesis 1 plus straight h right parenthesis squared minus 3 left parenthesis 1 plus straight h right parenthesis plus 3 over 2 close curly brackets space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below open curly brackets 2 straight h squared plus 4 straight h plus 2 minus 3 minus 3 straight h plus 3 over 2 close curly brackets equals 0 plus 0 plus 2 minus 3 minus 0 plus 3 over 2 equals 1 half space Also space straight f left parenthesis 1 right parenthesis equals Value space of space open parentheses straight x squared over 2 close parentheses space at space straight x equals 1 space space space space space space space space space space space space space space space space space equals 1 half therefore Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 1 right parenthesis

∴ f is continuous at x = 1.

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85. Is 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 x comma space if space straight x less or equal than 1 end cell row cell 5 comma space if space straight x greater than 1 end cell end table close

continous at x=0? At x=1? At x=2?

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Long Answer Type

86. Which of the following functions are continuous at the indicated points ?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space space space comma space if space straight x less than 0 end cell row cell straight x squared space space comma space if space straight x greater than 0 end cell end table close

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Short Answer Type

87. Which of the following functions are continuous at the indicated points?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space space 5 straight x minus 4 space space space comma space if space straight x less or equal than 1 end cell row cell 4 straight x squared minus 3 straight x space space comma space if space straight x greater than 1 end cell end table close

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88. Which of the following functions are continuous at the indicated points ?

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

71 Views

89. Which of the following functions are continuous at the indicated points ?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space straight x less than 0 end cell row cell 2 straight x plus 6 space space space comma space space space if space straight x greater than 1 end cell end table close at space straight x equals 0

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90. Which of the following functions are continuous at the indicated points ?

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

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Short Answer Type

Long Answer Type

Short Answer Type

84. Let f be defined byShow that f is continous at x=1.

Long Answer Type

Short Answer Type