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Continuity and Differentiability

Short Answer Type

71. Examine the following functions for continuity :

straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator straight x minus 5 end fraction

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72. Examine the following functions for continuity :

straight f left parenthesis straight x right parenthesis equals fraction numerator straight x squared minus 25 over denominator straight x minus 5 end fraction

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73. Examine the following functions for continuity :

f(x) = | x – 5 |

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74. Discuss the continuity of sine function

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75. Prove that the function defined by f(x) = tan x is a continuous function

Let f(x) = tan x

straight D subscript straight f equals straight R space except space od space multiples space of space straight pi over 2. Let space straight a space be space any space real space number space element of space straight D subscript straight f space space space space space space space space space space space space space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below tan space straight x space left square bracket Put space straight x equals straight a plus straight h 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 space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below space tan left parenthesis straight a plus straight h right parenthesis equals Lt with straight h rightwards arrow 0 below fraction numerator sin left parenthesis straight a plus straight h right parenthesis over denominator cos space left parenthesis straight a plus straight h right parenthesis end fraction 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 equals Lt with straight h rightwards arrow 0 below fraction numerator sin space straight a space cos space straight h plus cosa space sin space straight h over denominator cos space acos space straight h minus sin space straight a space sin space straight h end fraction 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 equals fraction numerator sin space straight a.1 plus cos space straight a.0 over denominator cos space straight a.1 minus sin space straight a.0 end fraction equals fraction numerator sin space straight a over denominator cos space straight a end fraction equals tan space straight a Also space space space space straight f left parenthesis straight a right parenthesis equals tan space straight a therefore space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight a right parenthesis equals tan space straight a

∴ f is continuous at x = a
But a is any member of D_f∴ f is continuous at every point of the domain,
∴ tan x is continuous at every point of the domain.

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Long Answer Type

76. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

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Short Answer Type

77. Discuss the continuity of the following functions :
(a) f(x) = sin x + cos x

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78. Discuss the continuity of the following functions :
–f(x) = sin x – cos x

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79. Discuss the continuity of the following functions :
f(x) = sin x.cos x

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80. Let f be a function such that f(x + y) =f(x) +f(y), x,y∈ R
Show that if f is continuous at x = 0, then it is continuous everywhere.

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