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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 =

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7.
Discuss the continuity of the function f given by f(x) = | x | at x = 0.

Here space space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x close vertical bar Lt with straight x rightwards arrow 0 to the power of minus below space space straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below space open vertical bar straight x close vertical bar space space space space space space space left square bracket Put space straight x equals 0 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 0 to the power of minus right square bracket space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below space open vertical bar 0 minus straight h close vertical bar equals Lt with straight h rightwards arrow 0 below open vertical bar negative straight h close vertical bar equals Lt with straight h rightwards arrow 0 below space straight h equals 0 Lt with straight x rightwards arrow 0 to the power of plus below space straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below space open vertical bar straight x close vertical bar space space space space space space space left square bracket Put space straight x equals 0 plus 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 0 plus right square bracket 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 open vertical bar 0 plus straight h close vertical bar equals Lt with straight h rightwards arrow 0 below space space open vertical bar straight h close vertical bar equals Lt with straight h rightwards arrow 0 below space straight h equals 0 Also space straight f space is space defined space at space straight x equals 0 and space space space space straight f left parenthesis 0 right parenthesis equals open vertical bar 0 close vertical bar equals 0 therefore space space Lt space straight f left parenthesis straight x right parenthesis equals Lt space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis therefore space space space straight f space is space continous space at space straight x equals 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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