Examine the continuity of f , where f is defined by from Mathem
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    Continuity and Differentiability

    Multiple Choice QuestionsShort Answer Type

    81. If f is continuous and g is a discontinuous function then f + g is discontinuous function.
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    Multiple Choice QuestionsLong Answer Type

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    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

    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
    Df = R
    Let a be any real number.
    Two cases arise :
    Case I : a ≠ 0
    space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below left parenthesis sin space straight x minus cos space straight x right parenthesis 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 Put space straight x equals straight a plus 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 straight a right square bracket 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 sin left parenthesis straight a plus straight h right parenthesis minus cos left parenthesis straight a plus straight h right parenthesis close curly brackets 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 left parenthesis sin space straight a space cos space straight h plus cos space straight a space cos space straight h right parenthesis minus left parenthesis cos space straight a space cos space straight h minus sin space straight a space sin space straight h right parenthesis close curly brackets space space space space space space space space space space space space space space space equals open curly brackets left parenthesis sin space straight a.1 plus cos space straight a.0 right parenthesis minus left parenthesis cos space straight a.1 minus sin space straight a.0 right parenthesis close curly brackets equals sin space straight a minus cos space straight a Also space straight f left parenthesis straight a right parenthesis equals sin space straight a minus cos space straight a therefore space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals sin space straight a minus cos space straight a
    ∴ f is continuous at x = a ≠  0.

    Case II : a = 0
    therefore space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below left parenthesis sin space straight x minus cos space straight x right parenthesis 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 space space space space space left square bracket Put space straight x equals 0 minus straight h comma straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 to the power of minus right square bracket space space 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 sin left parenthesis negative straight h right parenthesis minus cos left parenthesis negative straight h right parenthesis close curly brackets equals Lt with straight h rightwards arrow 0 below left parenthesis negative sin space straight h minus cos space straight h right parenthesis space space space space space space space space space space space space space space space space space space space space equals negative sin space 0 minus cos space 0 equals negative 0 minus 1 equals negative 1 space space space space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis sin space straight x minus space cos space straight x right parenthesis 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 space space space space left square bracket Put space straight x equals 0 plus 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 0 to the power of plus right square bracket space space 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 left parenthesis sin space straight h minus cos space straight h right parenthesis equals 0 minus 1 equals negative 1 Also space space space space space straight f left parenthesis 0 right parenthesis equals sin space 0 minus cos space 0 equals 0 minus 1 therefore space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis
    ∴ f is continuous at x = a = 0
    Thus, f is continuous at all points of its domain.

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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
    102 Views
    Answer

    Multiple Choice QuestionsShort 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.
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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?
    82 Views
    Answer

    Multiple Choice QuestionsLong 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
    71 Views
    Answer

    Multiple Choice QuestionsShort 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
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    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
    87 Views
    Answer
    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
    72 Views
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