Discuss the continuity of the following functions :–f(x) = sin
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    Continuity and Differentiability

    Multiple Choice QuestionsShort 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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    Answer
    73. Examine the following functions for continuity :

    f(x) = | x – 5 |

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    Answer
    74. Discuss the continuity of sine function
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    Answer
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    75. Prove that the function defined by f(x) = tan x is a continuous function
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    Answer

    Multiple Choice QuestionsLong Answer Type

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

    f(x) = sin x – cos x
    Df = R
    Let a be any real number.
    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 left parenthesis sin space straight x minus cos space straight x right parenthesis equals sin space straight a minus cos space straight a Also space straight f left parenthesis straight a right parenthesis equals sin space straight a minus cos 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 therefore space straight f space is space continuous space at space straight x equals straight a space But space straight a space is space any space point space of space straight R. space space therefore space straight f space is space continuous space function.

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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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