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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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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

g(x) = x – [x]
Let a be any integer.
. $Lt with straight x rightwards arrow straight a to the power of minus below straight g left parenthesis straight x space right parenthesis space equals Lt with straight x rightwards arrow straight a to the power of minus below space open curly brackets straight x minus left square bracket straight x right square bracket close curly brackets 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 space space space space space space left square bracket Put space straight x minus space straight a 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 straight a to the power of minus space space space space space space space space space space space space space space space space space space space equals stack Lt space space with straight x rightwards arrow 0 below space space open curly brackets left parenthesis straight a minus straight h right parenthesis minus left square bracket straight a minus straight h right square bracket close curly brackets equals space Lt with straight h rightwards arrow 0 below space open curly brackets left parenthesis straight a minus straight h right parenthesis minus left parenthesis straight a minus 1 right parenthesis close curly brackets 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 left parenthesis straight a minus straight h minus straight a plus 1 right parenthesis equals straight a minus 0 minus straight a plus 1 equals 1 Lt with straight x rightwards arrow straight a to the power of plus below space straight g left parenthesis straight x space right parenthesis equals space Lt with straight x rightwards arrow straight a to the power of plus below open curly brackets straight x minus left square bracket straight x right square bracket space close curly brackets 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 space space space left square bracket Put space straight x equals straight a 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 straight a to the power of plus right square bracket 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 space open curly brackets left parenthesis straight a plus straight h right parenthesis minus left square bracket straight a plus straight h right square bracket close curly brackets equals space Lt with straight h rightwards arrow 0 below space open curly brackets left parenthesis straight a plus straight h right parenthesis minus straight a close curly brackets equals space Lt with straight h rightwards arrow 0 below space left parenthesis straight h right parenthesis space space space space space space space space space space space space space space space space space space space equals 0 therefore space Lt with straight x rightwards arrow straight a to the power of minus below space straight f left parenthesis straight x right parenthesis not equal to space Lt with straight x rightwards arrow straight a to the power of plus below space straight f left parenthesis straight x right parenthesis$
∴ f is discontinuous at x = a
But a is any integral point.
∴ f is discontinuous at all integral points.

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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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