question_answer 3)

Examine continuity of the following functions.       (a) f(x) = x ? 5         (b) f(x)       (c) f(x) =     (d) f(x) = |x ? 5|  

Answer:

(a) f(x) = x ? 5 for x < 5 and x > 5, f(x) is a polynomial functions, so f(x) is continuous everywhere except possible at x = 5.        continuity at x = 5             = 5 ? 5 = 0       Also f(5) = 5 ? 5 = 0       Since        is continuous at x = 5.       Hence f(x) = x ? 5 is a continuous functions.       (b)is defined if x ? 5 0        is defined if x ? 5  0             for x < 5 and x > 5 f(x) is a rational functions.       Therefore f(x) is continuous function in its domain (Df)       (c)  is defined if x + 5             For x < ?5 and x > ?5, f(x) is a rational function. Therefore f(x) is a continuous function in its domain. (d)   f(x) = |x ? 5| For x < 5, f(x) = ? (x ? 5) and for x > 5,f(x) = x ? 5 (by definition) Therefore for x < 5 and x > 5, being a polynomial function, f(x) is continuous. Now continuity at x = 5                         = 0                         Also f(5) = |5 ? 5| = 0       Since LHL = RHL = f(5)        is continuous at x = 5.       Hence f(x) = |x ? 5| is a continuous function.