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TIFR-2019-Maths-A: 19

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Let $C^{\infty }\left ( 0,1 \right )$ stand for the set of all real-valued functions on $\left ( 0,1 \right )$ that have derivatives of all orders. Then the map $C^{\infty }\left ( 0,1 \right )\rightarrow C^{\infty }\left ( 0,1 \right )$ given by

$$f \mapsto f+\frac{df}{dx}$$

is

  1. injective but not surjective
  2. surjective but not injective
  3. neither injective nor surjective
  4. both injective and surjective

 

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