Question 59
Let $f$ be a real function of a real variable that checks the conditions
(i) $f(10+x)=f(10-x)$
(ii) $f(20+x)=-f(20-x)$
for each real value of $x$. Prove that $f$ is odd and periodic.
Remember that $f$ is said to be odd if $f(-x)=-f(x)$ for every $x$;
$f$ is said to be periodic if there exists $T\gt 0$ such that $f(x+T)=f(x)$ for every $x$.
(i) $f(10+x)=f(10-x)$
(ii) $f(20+x)=-f(20-x)$
for each real value of $x$. Prove that $f$ is odd and periodic.
Remember that $f$ is said to be odd if $f(-x)=-f(x)$ for every $x$;
$f$ is said to be periodic if there exists $T\gt 0$ such that $f(x+T)=f(x)$ for every $x$.