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TIFR-2016-Maths-A: 13

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Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function such that $\left | f\left ( x \right ) -f\left ( y \right )\right |\geq \left | x-y \right |$, for all $x,y\in\mathbb{R}$. Then the equation ${f}'\left ( x \right )=\frac{1}{2}$

  1. has exactly one solution 
  2. has no solution
  3. has a countably infinite number of solutions
  4. has uncountably many solutions
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