Statistical Cauchy and statistical boundedness of a function

Let D be a bounded interval in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℝ</m:mi> </m:math> {R} and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>→</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> </m:math> {f:D} be uniformly continuous on D . In this context, we can guarantee that f is bounded on D .” In this paper, implementing the concept of statistical Cauchy function, we show that uniform continuity is a sufficient condition for the boundedness of f on D . We too formulate a weaker condition rooted to the idea of statistical Cauchy property of f , that leads to the same conclusion.