Consider the following two statements:
$(E)$ Continuous function on $[1,2]$ can be approximated uniformly by a sequence of even polynomials (i.e., polynomials $p\left ( x \right )\in\mathbb{R}\left [ x \right ]$ such that $p\left ( -x \right )=p\left ( x \right )$).
$(O)$ Continuous function on $[1,2]$ can be approximated uniformly by a sequence of odd polynomials (i.e., polynomials $p\left ( x \right )\in\mathbb{R}\left [ x \right ]$ such that $p\left ( -x \right )=-p\left ( x \right )$).
Choose the correct option below.
- $(E)$ and $(O)$ are both false
- $(E)$ and $(O)$ are both true
- $(E)$ is true but $(O)$ is false
- $(E)$ is false but $(O)$ is true
