\title{Continuity of derivative}
\language{en}
\range{-0.9..0.9}
\author{Gang XIAO}
\email{xiao@unice.fr}
\computeanswer{yes}
\format{html}
\precision{10000}

\text{if=<i>f</i>}
\text{BR=<b>R</b>}
\text{fp=\if&nbsp;<tt>'</tt>}

\statement{Let \if: \BR -> \BR be a continuous function. If the derivative
\fp(x) exists for any point x&nbsp;<img
src=\imagedir/in.gif>&nbsp;\BR, is the derivative function
\fp: \BR -> \BR always continuous?
}

\choice{Reply}{No}{Yes}
\hint{Do you know the function \if such that \if(x)=x<sup>2</sup>sin(1/x)
when x&nbsp;<img src=\imagedir/ne.gif>&nbsp;0, and that
\if(0)&nbsp;=&nbsp;0&nbsp;?
<p>Compute the derivative \fp(0), and examine the limit of
\fp(x) when x goes towards 0.
}

