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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

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

Here space 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

The function f is defined at all points of the real line.
Let c be any real number
Case I: lf c < 1, then f(c) = c + 2.

therefore 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 straight f left parenthesis straight x plus 2 right parenthesis equals straight c plus 2 therefore 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 real numbers less than 1.

Case II: If c > 1, then f(c) = c – 2.
$therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis straight x minus 2 right parenthesis equals straight c minus 2 equals straight f left parenthesis straight c right parenthesis therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴f is continuous at all points x > 1.

Case III : If c = 1, then
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∴ f is not continuous at x = 1
∴ x = 1 is the only point of discontinuity of f.
∴ f is not a continuous function.