Increasing and decreasing piecewise function on an intervalI'm working on a problem that involves finding the intervals where a function f is increasing and decreasing. Given the function f ( x ) = { x + 7 text{if } xlt -3 | x + 1 | text{if } -3le x &lt;1 5 − 2 x text{if } xge 1 I worked out that f is increasing on ( − ∞ , − 3 ) and [ − 1 , 1 ), and f is decreasing on [ − 3 , − 1 ] , [ 1 , ∞ ).However, the solution in the book claims that f is increasing on [-1,1], rather than [-1,1) like I worked out. I am having trouble understanding why the book claims that this is the case.I was under the impression that f must be differentiable on the interior of an interval I and continuous on all of I in order to make any statements about increasing/decreasing behavior on the closed interval I. I'm not sure if I am overlooking something, but it seems that f is not continuous on [-1,1]. I would appreciate any clarification.

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Increasing and decreasing piecewise function on an intervalI&#039;m working on a problem that involves finding the intervals where a function f is increasing and decreasing. Given the function   f  (  x  )  =      {                            x          +          7                                       text{if } xlt -3                                                            |                    x          +          1                      |                                                 text{if } -3le x &amp;lt;1                                                5          −          2          x                                       text{if } xge 1                                  I worked out that f is increasing on   (  −  ∞  ,  −  3  ) and   [  −  1  ,  1  ), and f is decreasing on   [  −  3  ,  −  1  ]  ,  [  1  ,  ∞  ).However, the solution in the book claims that f is increasing on [-1,1], rather than [-1,1) like I worked out. I am having trouble understanding why the book claims that this is the case.I was under the impression that f must be differentiable on the interior of an interval I and continuous on all of I in order to make any statements about increasing/decreasing behavior on the closed interval I. I&#039;m not sure if I am overlooking something, but it seems that f is not continuous on [-1,1]. I would appreciate any clarification.

Alvin Preston · Accepted Answer

Step 1It is true that if you have a differentiable function on an interval, then it is increasing if and only if its derivative is non-negative. However, increasing functions need not be differentiable according to their definition:A function   f  :      R    →      R   is increasing on a collection S if and only if:For any   x  ,  y  ∈  S such that   x  ≤  y:  f  (  x  )  ≤  f  (  y  )Step 2Note that this definition is incompatible with the one that lulu proposed in a comment, but I believe this is the typical definition.

Deanna Gregory · Accepted Answer

Explanation:You have found that f is increasing on   [  −  1  ,  1  )Now       lim          x      →              1        −              f  (  x  )  =      |    1  +  1      |    =  2  &amp;lt;  3  =  f  (  1  ).Thus f is increasing on   [  −  1  ,  1  ]

Involves finding the intervals where a function f is increasing and decreasing. Given the function f(x)={(x+7,if x lt-3,,),(|x+1|,if -3 leq x<1,,),(5-2x,if x ge 1,,):}

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