Suppose that f , g : R → R are functions such that | f ( x ) − f ( y ) | ≤ | g ( x ) − g ( y ) | | x − y | for any x , y ∈ R. If g is differentiable with bounded derivative on all R, show that f is constant.I know I am meant to be using MVT for this question, I attempted to use g ( x ) as the function because we know it is differentiable (condition required for MVT): g ( b ) − g ( a ) b − a = g ′ ( c ) | g ( b ) − g ( a ) | = | g ′ ( c ) | b − a | I am lost, not sure where to start the question.

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Suppose that   f  ,  g  :  R  →  R are functions such that      |    f  (  x  )  −  f  (  y  )      |    ≤      |    g  (  x  )  −  g  (  y  )      |              |        x    −    y          |      for any   x  ,  y  ∈  R. If   g is differentiable with bounded derivative on all   R, show that   f is constant.I know I am meant to be using MVT for this question, I attempted to use   g  (  x  ) as the function because we know it is differentiable (condition required for MVT):            g      (      b      )      −      g      (      a      )              b      −      a        =      g    ′    (  c  )      |    g  (  b  )  −  g  (  a  )      |    =      |        g    ′    (  c  )      |    b  −  a      |  I am lost, not sure where to start the question.

Olive Guzman · Accepted Answer

If the derivative of   g is bounded as       |        g    ′    (  c  )      |    ≤  M for all   c, then      |                  f        (        x        )        −        f        (        y        )                    x        −        y              |    ≤                    |            g      (      x      )      −      g      (      y      )              |                            |            x      −      y              |              ≤  M            |        x    −    y          |      for any   x  ,  y, where the last inequality is from your application of the mean value theorem. Can you take it from here?

Suppose that f , g : R <mo stretchy="false">→ R are functions such th

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