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TIFR-2018-Maths-B: 7

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Let $A$ be the set of all continuous functions $f:\left [ 0,1 \right ]\rightarrow \left [ 0,\infty \right )$ satisfying the following condition:

$$\int_{0}^{x}f\left ( t \right )dt\geq f\left ( x \right ), \:for\:all \:x\in\left [ 0,1 \right ].$$

Then which of the following statements is true?

  1. $A$ has cardinality $1$.
  2. $A$ has cardinality $2$.
  3. $A$ is infinite.
  4. $A$ is empty.
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