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Continuity and Differentiability

Short Answer Type

21.
Examine the continuity of
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x minus straight a close vertical bar over denominator straight x minus straight a end fraction space space space space space space straight x not equal to straight a end cell row cell space space space space space space space space 1 space space space space space space space space space space space straight x equals straight a end cell end table close space at space space straight x equals straight a$

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x minus straight a close vertical bar over denominator straight x minus straight a end fraction space space space space space space straight x not equal to straight a end cell row cell space space space space space space space space 1 space space space space space space space space space space space straight x equals straight a end cell end table close space Lt with straight x rightwards arrow straight a to the power of minus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight a to the power of minus below fraction numerator open vertical bar straight x minus straight a close vertical bar over denominator straight x minus straight a end fraction space left square bracket Put space straight x equals straight a minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a to the power of minus right square bracket space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar straight a minus straight h minus straight a close vertical bar over denominator straight a minus straight h minus straight a end fraction equals space Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar negative straight h close vertical bar over denominator negative straight h end fraction equals space Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator negative straight h end fraction equals negative 1 space Lt with straight x rightwards arrow straight a to the power of plus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight a to the power of plus below fraction numerator open vertical bar straight x minus straight a close vertical bar over denominator straight x minus straight a end fraction space left square bracket Put space straight x equals straight a plus straight h comma space straight h greater than space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a right square bracket space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar straight a plus straight h minus straight a close vertical bar over denominator straight a plus straight h minus straight a end fraction equals space Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar straight h close vertical bar over denominator straight h end fraction equals space Lt with straight h rightwards arrow 0 below straight h over straight h equals 1 Also space straight f left parenthesis straight a right parenthesis equals 1 therefore space Lt with straight x rightwards arrow straight a to the power of minus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight a to the power of plus below space

∴ f is continuous at x = a.

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22. Examine the continuity of f (x) at x = 0.

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator 2 open vertical bar straight x close vertical bar end fraction comma space straight x not equal to 0 end cell row cell space space space space space 1 half comma space straight x equals 0 end cell end table close

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23. Examine the continuity of f (x) at x = 0.

If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus open vertical bar straight x close vertical bar comma space straight x not equal to 0 end cell row cell space space space space space space 2 space space space comma space straight x equals 0 end cell end table close

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24. Test the continuity of the function f(x) at the origin :

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma space space space space straight x not equal to 0 end cell row cell space space space 1 space space space space comma space space space space straight x equals 0 end cell end table close

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Long Answer Type

25. Test the continuity of the function f(x) at the origin

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma space space space space straight x not equal to 0 end cell row cell space space space 0 space space space space comma space space space space straight x equals 0 end cell end table close

then show that f(x) is discontinuous at x = 0.

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26. Prove that the function

straight f left parenthesis straight x right parenthesis open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar plus 2 straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space straight k space space space space space space space comma space straight x equals 0 end cell end table close

remains discontinuous at x = 0, regardless of the choice of k.

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27. Show that the function f given by f(x) = | x | + | x –1 |,x ∈ R is continuous both at x =.0 and x = 1.

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28.

Discuss continuity of the function f given by

f(x) = | x – 1| + | x – 2 ] at x = 1 and x = 2.

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Short Answer Type

29. Discuss the continuity or otherwise of the function f(x) defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin 3 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space 1 space space space space space space comma space straight x equals 0 end cell end table close space at space straight x equals 0

87 Views

30. Discuss the continuity or otherwise of the function f(x) defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 2 straight x over denominator sin space 3 straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space 2 space space space space space comma space straight x equals 0 end cell end table close at space straight x equals 0

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21. Examine the continuity of

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