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TIFR-2016-Maths-A: 5

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Which of the following continuous functions $f:\left ( 0,\infty \right ) \rightarrow \mathbb{R}$ can be extended to a continuous function on $\left [ 0,\infty \right )$ ?

  1. $f\left ( x \right )=sin\frac{1}{x}$
  2. $f\left ( x \right )=\frac{1-cos\:x}{x^{2}}$
  3. $f\left ( x \right )=cos\frac{1}{x}$
  4. $f\left ( x \right )=\frac{1}{x}$
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