Let us consider the identity function f : ( R , d ) → ( R , d u s u a l ) f : x → xHere we are considering d ( x , y ) = | ( x ) 3 − ( y ) 3 | Is the function f uniformly continuous on closed and bounded interval?I am looking for an example of function f which is not uniformly continuous on a closed and bounded interval but it is continuous.

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Let us consider the identity function  f  :  (      R    ,  d  )  →  (      R    ,      d          u      s      u      a      l        )  f  :  x  →  xHere we are considering   d  (  x  ,  y  )  =      |    (  x      )    3    −  (  y      )    3        |  Is the function   f uniformly continuous on closed and bounded interval?I am looking for an example of function   f which is not uniformly continuous on a closed and bounded interval but it is continuous.

ohhappyday890b · Accepted Answer

In this answer I assume that       tan          −      1       denotes the arctangent arctan.No, your example does not work, because arctan is Lipschitz continuous. Indeed for all   x  ,  y  ∈      R        |    arctan  ⁡  x  −  arctan  ⁡  y      |    ≤      |    x  −  y      |  this implies that the identity map   f  :      R    →  (      R    ,  d  ) is Lipschitz (hence uniformly continuous).Anyway, if you want to build an example of a continuous function which is not uniformly continous on a closed and bounded interval, you are trying to find a counterexample to Heine-Cantor theorem. In particular you need to find a distance defined on the real line with the property that closed and bounded intervals are not compact.

Let us consider the identity function f:(R,d)->(R,d_usual) f:x->x Here we are considering d(x,y)=|(x)^3−(y)^3| Is the function f uniformly continuous on closed and bounded interval?

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