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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.

Let f(x) =| x |, g (x) =sin x
D_f = R, R_f =[ 0, ∞ )
Dg =R, R_g=[–1,1]
∵ R_f. C Dg
∵ g o f is defined
and ( g o f) (x) =g(f(x)) =g(| x |)=sin| x |
Now f and g are both continuous for every x ∈ R.
∴ g o f i.e., sin | x | is continuous for every x ∈ R.

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