 By Klymchuk S.

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Additional resources for Counter-examples in calculus

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The function below is defined on [a,b] and continuous on (a,b) but for any N  ( f (a ), f (b )) there is no point c  (a , b ) such that f(c) = N. f(b) f(a) a b 53 15. If a function is discontinuous at every point in its domain then the square and the absolute value of this function cannot be continuous. Counter-example. The function f ( x ) ­ 1, if x is rational is discontinuous at every ® ¯  1, if x is irrational point in its domain but both the square and the absolute value f 2 ( x ) f ( x ) 1 are continuous.

0 the following is true: if x < a then f(x) < f(a) and if x > a then f(x) > f(a). 14. 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.

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. y x3 2 8 -6 -4 0 -2 0 2 4 6 8 -2 4 Comments. To make the statement true it should conclude: “…then its absolute value f ( x ) is not differentiable at the points where f(x) = 0 and f c( x ) z 0.