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TIFR-2018-Maths-B: 14

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Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Then which of the following statements implies that $f\left ( 0 \right )=0$?

  1. $\underset{n\rightarrow \infty }{lim}\int_{0}^{1}f\left ( x \right )^{n}dx=0.$
  2. $\underset{n\rightarrow \infty }{lim}\int_{0}^{1}f\left ( x/n\right )dx=0.$
  3. $\underset{n\rightarrow \infty }{lim}\int_{0}^{1}f\left ( nx\right )dx=0.$
  4. None of the above.
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