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TIFR-2014-Maths-A-4

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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

  1. $f$ is discontinuous at $x = 0$ 
  2. $f$ is continuous on $[0, \infty)$, but not uniformly continuous on $[0, \infty)$ 
  3. $f$ is uniformly continuous on $[0, \infty)$ 
  4. $f$ is not uniformly continuous on $[0, \infty)$, but uniformly continuous on $(0, \infty)$
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