Let $f$ be the real valued function on $[0, \infty)$ defined by
$f(x)
= \begin{cases}
x^{\frac{2}{3}}\log x& \text {for x > 0} \\
0& \text{if x=0 }
\end{cases}$
Then
- $f$ is discontinuous at $x = 0$
- $f$ is continuous on $[0, \infty)$, but not uniformly continuous on $[0, \infty)$
- $f$ is uniformly continuous on $[0, \infty)$
- $f$ is not uniformly continuous on $[0, \infty)$, but uniformly continuous on $(0, \infty)$.