# Counter-examples in calculus by Klymchuk S.

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By Klymchuk S.

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Counter-example. The function below is a one-to-one function and has an inverse function on (a,b) but it is neither increasing nor decreasing on (a,b). 30 a 8. b A function y = f(x) is bounded on R if for any x  R there is M > 0 such that f ( x ) d M . Counter-example. For the function y x 2 for any value of x from R there is a number M > 0 ( M x 2  H , where H t 0 ) such that f ( x ) d M . Comments. The order of words in this statement is very important. The correct definition of a function bounded on R differs only by the order of words: A function y = f(x) is bounded on R if there is M > 0 such that for any x  R f ( x ) d M .

Takes all its values between the maximum and minimum values; then this function is continuous on [a,b]. Counter-example. The function below satisfies the three conditions above, but is not continuous on [a,b]. a b 11. If on the closed interval [a,b] a function is: a. bounded; b. takes its maximum and minimum values; c. takes all its values between the maximum and minimum values; then this function is continuous at some points or subintervals on [a,b]. Counter-example. The function below satisfies all three conditions above but it is discontinuous at every point on [-1,1].

47 Comments. A strict inequality in the definition of a local maximum is accepted here: a function y = f(x) has a local maximum at the point x = a if f (a ) ! f ( x ) for all x within a certain neighbourhood (a  G , a  G ), G ! 0 of the point x = a. Otherwise in the above graph we have to treat each point of the line segment as a local maximum. 6. If a function is defined in a certain neighbourhood of point x = a including the point itself and is increasing on the left from x = a and decreasing on the right from x = a, then there is a local maximum at x = a.