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Continuity and Differentiability

Short Answer Type

1. Check the continuity of the function f given by f(x) = 2 x + 3 at x = 1.

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Long Answer Type

2. Prove that the function f(x) = 5 x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

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Short Answer Type

3. Examine the continuity of the function f(x) = 2x² –1 at.x = 3

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4. Prove that the function f(x) = xⁿ is continuous at x = n, where n is a positive integer.

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5. Examine whether the function f given by f(x) = x² is continuous at x = 0.

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6. Is the function defined by f(x) = x² –sin x + 5 continuous at x =?

straight f left parenthesis straight x right parenthesis equals straight x squared minus space sinx plus 5 Lt with straight x rightwards arrow straight pi below space straight f left parenthesis straight x right parenthesis equals stack space Lt space with straight x rightwards arrow straight pi below left parenthesis straight x squared minus space sinx space plus 5 right parenthesis 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 space space space space space equals straight pi squared minus space sinπ plus 5 equals straight pi squared minus 0 plus 5 equals straight pi squared plus 5 Also space straight f left parenthesis straight pi right parenthesis space equals straight pi squared minus space sinπ plus 5 equals straight pi squared minus 0 plus 5 equals straight pi squared plus 5 therefore space Lt with straight x rightwards arrow straight pi below space straight f left parenthesis straight pi right parenthesis equals straight f left parenthesis straight pi right parenthesis therefore space space straight f space is space continous space at space straight x equals straight pi.

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Discuss the continuity of the function f given by f(x) = | x | at x = 0.

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8. Prove that the identity function on real numbers given by f(x) = x is continuous at every real number.

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9. Show that the function defined by g(x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x

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10. Find all the points of discontinuity of the greatest integer function defined by f(x) = [x], where [x] denotes the greatest integer less than or equal to x.

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