23. Show that if f x is continuous on bounded a; b, so too is j f...

23. Show that if f ðxÞ is continuous on bounded ½a; b, so too is j f ðxÞj. In other words, show that f ðxÞ ! f ðx0Þ implies that j f ðxÞj ! j f ðx0Þj. (Hint: To show this, prove that j jaj  jbj jaja  bj:Þ

(a) Give a di¤erent example from what is in the text of where j f ðxÞj continuous does not imply that f ðxÞ is continuous.

(b) Give a second example where the continuity of f ðxÞ2 does not imply the continuity of f ðxÞ.