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If a function is not monotone then it doesn’t have an inverse function. Counter-example. The function y x , if x is rational is not monotone but it has ® ¯ x , if x is irrational 34 the inverse function x y , if y is rational ® ¯ y , if y is irrational. It is impossible to draw the graph of such a function but a rough sketch gives an idea of its behaviour: y x , if x is rational ® ¯ x , if x is irrational 15. If a function is not monotone on (a,b) then its square cannot be monotone on (a,b).
There are infinitely many values of the function y 1 1 sin x x y 2 8 -6 -4 -2 0 0 2 -2 Comments. The correct definition is a). 39 4 6 8 5. If lim f ( x ) exists and lim g ( x ) does’t exist because of oscillation of xo a xo a g(x) near x = a then lim( f ( x ) u g( x )) doesn’t exist. x oa Counter-example. For the function f(x) = x the limit lim x xo0 0 and for the function 1 1 the limit lim sin doesn’t exist because of oscillation of x o 0 x x 1 g(x) near x = 0, but lim( f ( x ) u g ( x )) lim( x sin ) 0 .
If a function y = f(x) is differentiable for all real x and f (0) f c(0) 0 then f(x) = 0 for all real x. Counter-example. Both the function y x 2 and its derivative y c 2 x equal zero at the point x = 0 but the function is not zero for all real x. 62 4 y x2 2 8 -6 -4 0 -2 0 2 4 6 8 -2 -4 12. g. f ( x) x or f ( x ) sin x . Counter-example. x 3 is differentiable on R and takes both positive The function y x 3 is differentiable and negative values but its absolute value y at the point x = 0 where the function equals zero.