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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

∴ f is continuous function
∴ f is continuous at x = 2 and x = 10
∴ f is right continuous at x = 2 and left continuous at .x = 10.

therefore space space Lt with straight x rightwards arrow 2 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 2 right parenthesis space rightwards double arrow space 2 straight a plus straight b equals 5 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 parenthesis 1 right parenthesis space space space space space Lt with straight x rightwards arrow 10 to the power of minus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 10 right parenthesis space rightwards double arrow space 10 straight a plus straight b equals 21 space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

Subtracting (1) from (2), we get,
8 a = 16 or a = 2
∴ from (1), 4 + 6 = 5⇒ b = 1
∴ we have a = 2, b = 1

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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|.

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128. If f is differentiable at r = a, then prove that

Lt with straight x rightwards arrow straight a below fraction numerator straight x space straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals straight f left parenthesis straight a right parenthesis equals straight a space straight f left parenthesis straight a right parenthesis

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129. Let a function f r be continuous at a. Show that the function g (x) = (x - a)f(x) is differentiable at x = a.

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130. Let f be differentiable at a, then evaluate

Lt with straight x rightwards arrow straight a below fraction numerator straight x squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction

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