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Continuity and Differentiability

Short Answer Type

101. Find the value of k so that the function f 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 space straight k space straight x squared comma space straight x greater or equal than 1 end cell row cell space 4 space space space space comma space straight x less than 1 end cell end table close at space straight x equals 1

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102. Find the value of k so that the function f 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 squared comma space straight x less or equal than 2 end cell row cell 3 space space space space comma space straight x greater than 2 end cell end table close at space straight x equals 2

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103. Find the value of k so that the function f 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 1 space space comma space straight x less or equal than 5 end cell row cell 3 x minus 5 space space space comma space straight x greater than 5 end cell end table close at space straight x equals 5

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104. Find the value of k so that the function f 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 left parenthesis straight x squared minus 2 straight x space right parenthesis comma space if space straight x less than 0 end cell row cell space space cos space straight x space space space space space comma space if space straight x greater or equal than 0 end cell end table close at space straight x equals 0

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105. Discuss the continuity of 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 plus 2 comma space if space straight x less or equal than 1 end cell row cell straight x minus 2 comma space if space straight x greater than 1 end cell end table close

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106. Find all the points of discontinuity of the function f defined by

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

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107. Discuss the continuity of 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 x plus 2 comma space space space space space if space straight x less than 0 end cell row cell negative straight x plus 2 comma space if space straight x greater than 0 end cell end table close

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108. 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 space space straight x comma space if space straight x greater or equal than 0 end cell row cell straight x squared comma space if space straight x less than 0 end cell end table close

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109. Find all points of discontinuity 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 2 straight x plus 3 comma space if space straight x less or equal than 2 end cell row cell 2 straight x minus 3 comma space if space straight x greater than 2 end cell end table close

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Long Answer Type

110. Find all points of discontinuity of f, where f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell open vertical bar straight x close vertical bar plus 3 comma space if space straight x less or equal than negative 3 end cell row cell negative 2 straight x comma if space minus 3 less than straight x greater than 3 end cell row cell 6 straight x plus 2 space space comma space if space straight x greater or equal than 3 end cell end table close$

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell open vertical bar straight x close vertical bar plus 3 comma space if space straight x less or equal than negative 3 end cell row cell negative 2 straight x comma if space minus 3 less than straight x greater than 3 end cell row cell 6 straight x plus 2 space space comma space if space straight x greater or equal than 3 end cell end table close

Function f is defined for all points of the real line.
Let c be any real number.
Five cases arise :
Case I: c < –3

$space space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis open vertical bar straight x close vertical bar plus 3 right parenthesis equals open vertical bar straight c close vertical bar plus 3 Also space space space space space space space straight f left parenthesis straight c right parenthesis equals open vertical bar straight c close vertical bar plus 3 therefore space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f is continuous at all point x < – 3
Case II: c = –3

$Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of minus below left parenthesis open vertical bar straight x close vertical bar plus 3 right parenthesis equals open vertical bar negative 3 close vertical bar plus 3 equals 3 plus 3 equals 6 Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below left parenthesis negative 2 straight x right parenthesis equals 6 Also space straight f left parenthesis straight c right parenthesis equals straight f left parenthesis negative 3 right parenthesis equals open vertical bar negative 3 close vertical bar plus 3 equals 3 plus 3 plus equals 6 therefore space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis equals 6$
∴ f is continuous at x = – 3
Case III: – 3 < c < 3
f(x) = – 2 x is a continuous function as it is a polynomial.
Case IV : c = 3

space space space space space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of minus below left parenthesis negative 2 straight x right parenthesis equals negative 6 space space space space space Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below left parenthesis 6 straight x plus 2 right parenthesis equals 18 plus 2 equals 20 therefore space space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis

∴ f is discontinuous at x = 3.
Case V ; c > 3

$space space space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis 6 straight x plus 2 right parenthesis equals 6 straight c plus 2 Also space space space space space space space straight f left parenthesis straight c right parenthesis equals 6 straight c plus 2 therefore space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f is continuous at all points x > 3.

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