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TIFR-2014-Maths-A-17

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Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function and let $S$ be a non-empty proper subset of $R$. Which one of the following statements is always true? (Here $\bar{A}$ denotes the closure of $A$ and $A^{∘}$ denotes the interior of $A$). 

  1. $f(S)^{∘} \subseteq f(S^{∘})$
  2. $f(\bar{S}) \subseteq \overline{f(S)}$ 
  3. $f(\bar{S}) \supseteq \overline{f(S)}$ 
  4. $f(S)^{∘} \supseteq f(S^{∘})$.
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