Question: Sketch all the continuous functions f : R → R which satisfy ( f ( x ) ) 2 = ( x − 1 ) 2 ( x − 2 ) 2 .Justify your answers.I have found the eight possible continuous functions as follows: f ( x ) = ( x − 1 ) ( x − 2 ) f ( x ) = − ( x − 1 ) ( x − 2 ) f ( x ) = | x − 1 | ( x − 2 ) f ( x ) = − | x − 1 | ( x − 2 ) f ( x ) = ( x − 1 ) | x − 2 | f ( x ) = − ( x − 1 ) | x − 2 | f ( x ) = | x − 1 | | x − 2 | f ( x ) = − | x − 1 | | x − 2 | . and have got the conclusion of this question which has at most eight possible such functions, but how to proof the "at most" statement.

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Question: Sketch all the continuous functions   f  :      R    →      R   which satisfy  (  f  (  x  )      )    2    =  (  x  −  1      )    2    (  x  −  2      )    2    .Justify your answers.I have found the eight possible continuous functions as follows:                    f        (        x        )                            =        (        x        −        1        )        (        x        −        2        )                            f        (        x        )                            =        −        (        x        −        1        )        (        x        −        2        )                            f        (        x        )                            =                  |                x        −        1                  |                (        x        −        2        )                            f        (        x        )                            =        −                  |                x        −        1                  |                (        x        −        2        )                            f        (        x        )                            =        (        x        −        1        )                  |                x        −        2                  |                                    f        (        x        )                            =        −        (        x        −        1        )                  |                x        −        2                  |                                    f        (        x        )                            =                  |                x        −        1                  |                          |                x        −        2                  |                                    f        (        x        )                            =        −                  |                x        −        1                  |                          |                x        −        2                  |                .            and have got the conclusion of this question which has at most eight possible such functions, but how to proof the &quot;at most&quot; statement.

Abagail Stephenson · Accepted Answer

Any such function   f has to vanish at   x  =  1 and   x  =  2 and nowhere else. Therefore the sign of   f has to be constant in each of the intervals       ]      −    ∞    ,  1  [    ,       ]  1  ,  2  [    ,       ]  2  ,  ∞  [   . It follows that you have exactly two choices for the graph of   f in each of these intervals. As the value at the junctions is 0 it follows that   f will automatically be continuous for any choice made. In particular you are allowed to make the choices independently. This gives a total of       2    3    =  8 possibilities.

Sketch all the continuous functions f:R->R which satisfy (f(x))^2=(x−1)^2(x−2)^2. Justify your answers.

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