Question: Let V be the real vector space of all functions \(f:\mathbb{R}\to \mathbb{R}\) Which of the following subsets of V are subspaces?

1. \(\left\{ f\in V:\,f\left( {{x}^{2}} \right)={{\left( f\left( x \right) \right)}^{2}}\text{ for all }x \right\}\)
2. \(\left\{ f\in V:\,f\left( 0 \right)=f\left( 1 \right) \right\}\)
3. \(\left\{ f\in V:\,f\left( 3 \right)=1+f\left( -5 \right) \right\}\)
4. \(\left\{ f\in V:\,\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)\text{=f}\left( a \right)\text{ }\,\text{for all a} \right\}\)

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