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Continuity and Differentiability

Short Answer Type

121.

Find the values of a and b such that the function defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 5 comma end cell row cell ax plus straight b comma end cell row cell 21 comma end cell end table close table row cell if space straight x less or equal than 2 space space space space space space space space space space end cell row cell if space 2 less than straight x less than 10 end cell row cell if space straight x greater or equal than 10 space space space space space space space space end cell end table$
is a continous function.

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122. Show that the function defined by f(x) = sin (x² ) is a continuous function.

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123. Show that the function defined by f(x) = cos(x²) is a continuous function.

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124. Examine that sin | x | is a continuous function.

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125. Show that the function defined by f(x) = | cos x | is a continuous function.

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126. Show that the function f defined by f(x) = |1–x+|x||, where x is any real number, is a continuous function.

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127.
Find all the points of discontinuity of f defined by
f(x) =|x|–|x+1|.

Here f(x) =|x|–|x+1|
Let g(x) = |x| and h(x) = x+1
∴ (g o h)(x) = g(h(x)) = g(x+1) = |x+1|
Now g and h are both continuous functions
∴ (g o h) is a continuous function.
∴ |x+1| is continuous function
Also | x | is continuous function.
Now difference of two continuous functions is a continuous function.
∴ f(x) = |x|–|x+1| is continuous.

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