[12 points] Throughout this problem we are interested in real valued functions $f$ satisfying two conditions: at each $x$ in its domain, $f$ is continuous and $f(x^{2}) = f(x)^{2}$. Prove the following independent statements about such functions. The hints below may be useful. (i) There is a unique such function $f$ with domain $[0,1]$ and $f(0) \neq 0$. (ii) If the domain of such $f$ is $(0, \infty)$, then ($f(x) = 0$ for every $x$) OR ($f(x) \neq 0$ for every $x$). (iii) There are infinitely many such $f$ with domain $(0, \infty)$ such that $\int_{0}^{\infty} f(x)\, dx < 1$. Hints: (1) Suppose a number $a$ and a sequence $x_{n}$ are in the domain of a continuous function $f$ and $x_{n}$ converges to $a$. Then $f(x_{n})$ must converge to $f(a)$. For example $f(0.5^{n}) \rightarrow f(0)$ and $f(2^{\frac{1}{n}}) \rightarrow f(1)$ if all the mentioned points are in the domain of $f$. In parts (i) and (ii) suitable sequences may be useful. (2) Notice that $f(x) = x^{r}$ satisfies $f(x^{2}) = f(x)^{2}$.
[12 points] Throughout this problem we are interested in real valued functions $f$ satisfying two conditions: at each $x$ in its domain, $f$ is continuous and $f(x^{2}) = f(x)^{2}$. Prove the following independent statements about such functions. The hints below may be useful.\\
(i) There is a unique such function $f$ with domain $[0,1]$ and $f(0) \neq 0$.\\
(ii) If the domain of such $f$ is $(0, \infty)$, then ($f(x) = 0$ for every $x$) OR ($f(x) \neq 0$ for every $x$).\\
(iii) There are infinitely many such $f$ with domain $(0, \infty)$ such that $\int_{0}^{\infty} f(x)\, dx < 1$.
Hints: (1) Suppose a number $a$ and a sequence $x_{n}$ are in the domain of a continuous function $f$ and $x_{n}$ converges to $a$. Then $f(x_{n})$ must converge to $f(a)$. For example $f(0.5^{n}) \rightarrow f(0)$ and $f(2^{\frac{1}{n}}) \rightarrow f(1)$ if all the mentioned points are in the domain of $f$. In parts (i) and (ii) suitable sequences may be useful. (2) Notice that $f(x) = x^{r}$ satisfies $f(x^{2}) = f(x)^{2}$.