![\"\"](\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2021\/09\/rd-sharma-solutions-for-class-12-maths-chapter-9-continuity-image-1.gif\" "\\"RD")
<\/figure>\n\n\n\nSolution:<\/strong><\/p>\n\n\n\n Given<\/p>\n\n\n\n Consider LHL at x = 0<\/p>\n\n\n\n 2. A function f(x) is defined as<\/strong><\/p>\n\n\n\n Show that f(x) is continuous at x = 3.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Given<\/p>\n\n\n\n 3. A function f(x) is defined as<\/strong><\/p>\n\n\n\n Show that f(x) is continuous at x = 3.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Find whether f(x) is continuous at x = 1.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Find whether f(x) is continuous at x = 0.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Find whether f(x) is continuous at x = 0.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Given<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Given<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 10. Discuss the continuity of the following functions at the indicated point(s):<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Given<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 13. Find the value of a for which the function f defined by<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 14. Examine the continuity of the function<\/strong><\/p>\n\n\n\n Also sketch the graph of this function.<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Hence f (x) is discontinuous at x = 0<\/p>\n\n\n\n 15. Discuss the continuity of the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 16. Discuss the continuity of the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 17. Discuss the continuity of<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 18. For what value of k is the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 19. Determine the value of the constant k so that the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 20. For what value of k is the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 21. Determine the value of the constant k so that the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 22. Determine the value of the constant k so that the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n 23. Find the values of a so that the function<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n Exercise 9.2 Page No: 9.34<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n A real function f is said to be continuous at x = c, where c is any point in the domain of f if<\/p>\n\n\n\n Where h is a very small positive number. i.e. left hand limit as x \u2192 c (LHL) = right hand limit as x \u2192 c (RHL) = value of function at x = c.<\/p>\n\n\n\n A function is continuous at x = c if<\/p>\n\n\n\n To prove it everywhere continuous we need to show that at every point in the domain of f(x) [domain is nothing but a set of real numbers for which function is defined]<\/p>\n\n\n\n Clearly from definition of f(x), f(x) is defined for all real numbers.<\/p>\n\n\n\n Now we need to check continuity for all real numbers.<\/p>\n\n\n\n Let c is any random number such that c < 0 [thus c being a random number, it can include all negative numbers]<\/p>\n\n\n\n We can say that f(x) is continuous for all x < 0<\/p>\n\n\n\n Now, let m be any random number from the domain of f such that m > 0<\/p>\n\n\n\n Thus m being a random number, it can include all positive numbers]<\/p>\n\n\n\n Therefore we can say that f(x) is continuous for all x > 0<\/p>\n\n\n\n As zero is a point at which function is changing its nature so we need to check LHL, RHL separately<\/p>\n\n\n\n f (0) = 0 + 1 = 1 [using equation 1]<\/p>\n\n\n\n Thus LHL = RHL = f (0).<\/p>\n\n\n\n Therefore f (x) is continuous at x = 0<\/p>\n\n\n\n Hence, we proved that f is continuous for x < 0; x > 0 and x = 0<\/p>\n\n\n\n Thus f(x) is continuous everywhere.<\/p>\n\n\n\n Hence, proved.<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n A real function f is said to be continuous at x = c, where c is any point in the domain of f if<\/p>\n\n\n\n Where h is a very small positive number. i.e. left hand limit as x \u2192 c (LHL) = right hand limit as x \u2192 c (RHL) = value of function at x = c.<\/p>\n\n\n\n A function is continuous at x = c if<\/p>\n\n\n\n Here we have,<\/p>\n\n\n\n \u2026\u2026.equation 1<\/p>\n\n\n\n The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)<\/p>\n\n\n\n Function is changing its nature (or expression) at x = 0, so we need to check its continuity at x = 0 first.<\/p>\n\n\n\n We know that from the definition of mod function we have<\/p>\n\n\n\n f (0) = 0 [using equation 1]<\/p>\n\n\n\n Clearly, LHL \u2260 RHL \u2260 f (0)<\/p>\n\n\n\n \u2234 Function is discontinuous at x = 0<\/p>\n\n\n\n Let c be any real number such that c > 0<\/p>\n\n\n\n Therefore f (x) is continuous everywhere for x > 0.<\/p>\n\n\n\n Let c be any real number such that c < 0<\/p>\n\n\n\n Therefore f (c) = [Using equation 1 and idea of mod function]<\/p>\n\n\n\n Therefore f (x) is continuous everywhere for x < 0.<\/p>\n\n\n\n Hence, we can conclude by stating that f (x) is continuous for all Real numbers except zero that is discontinuous at x = 0.<\/p>\n\n\n\n 3. Find the points of discontinuity, if any, of the following functions:<\/strong><\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n A real function f is said to be continuous at x = c, where c is any point in the domain of f if<\/p>\n\n\n\n Where h is a very small positive number. i.e. left hand limit as x \u2192 c (LHL) = right hand limit as x \u2192 c (RHL) = value of function at x = c.<\/p>\n\n\n\n A function is continuous at x = c if<\/p>\n\n\n\n Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)<\/p>\n\n\n\n Function is changing its nature (or expression) at x = 1, so we need to check its continuity at x = 1.<\/p>\n\n\n\n Clearly, f (1) = 4 [using equation 1]<\/p>\n\n\n\n \u2234 f (x) is discontinuous at x = 1.<\/p>\n\n\n\n Let c be any real number such that c \u2260 0<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n A real function f is said to be continuous at x = c, where c is any point in the domain of f if<\/p>\n\n\n\n Where h is a very small positive number. i.e. left hand limit as x \u2192 c (LHL) = right hand limit as x \u2192 c (RHL) = value of function at x = c.<\/p>\n\n\n\n A function is continuous at x = c if<\/p>\n\n\n\n The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)<\/p>\n\n\n\n Function is changing its nature (or expression) at x = 2, so we need to check its continuity at x = 2 first.<\/p>\n\n\n\n Clearly, f (2) = 16 [from equation 1]<\/p>\n\n\n\n \u2234 f (x) is continuous at x = 2.<\/p>\n\n\n\n Let c be any real number such that c \u2260 0<\/p>\n\n\n\n Solution:<\/strong><\/p>\n\n\n\n A real function f is said to be continuous at x = c, where c is any point in the domain of f if<\/p>\n\n\n\n\n\n Where h is a very small positive number. i.e. left hand limit as x \u2192 c (LHL) = right hand limit as x \u2192 c (RHL) = value of function at x = c.<\/p>\n\n\n\n![\"\"](\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2021\/09\/rd-sharma-solutions-for-class-12-maths-chapter-9-continuity-image-2.gif\" "\\"RD")
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