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

For f to be defined,
x + 5 ≠ 0, i.e. x ≠ – 5
∴D_f = Set of all real numbers except – 5 = R - {- 5}
Let c≠– 5 be any real number.

therefore space straight f left parenthesis straight c right parenthesis equals fraction numerator straight c squared minus 25 over denominator straight c plus 25 end fraction equals fraction numerator left parenthesis straight c minus 5 right parenthesis left parenthesis straight c plus 5 right parenthesis over denominator straight c plus 5 end fraction equals straight c minus 5 Also space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below fraction numerator straight x squared minus 25 over denominator straight x plus 5 end fraction equals Lt with straight x rightwards arrow straight c below fraction numerator left parenthesis straight x minus 5 right parenthesis left parenthesis straight x plus 5 right parenthesis over denominator straight x plus 5 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 equals Lt with straight x rightwards arrow straight c below left parenthesis straight x minus 5 right parenthesis equals straight c minus 5 therefore space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at x = c.
But c ≠ – 5 is any real number.
∴ f is continuous at every real number c ∈ D_f.
∴ f is continuous function.

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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

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

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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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