Calculate the modulus of uniform continuity of the functions in $R$,
a) $x→ sin (\frac{1}{x})$ para $ x>0$,
b)$ x→sin (x^2)$,
c) $x→x^2$
Where the modulus of uniform continuity of a function $ f:A→R $ es:
$φf(δ):=sup{|f(x)−f(y)|:x,y∈A,|x−y|≤δ}$
for the first function I have solved it like this:
$x=\frac{1}{\frac{π}{2}+2kπ}$, $f(x)=1$ y en y $=\frac{1}{-\frac{π}{2}+2kπ}$, $f(y)=−1$.therefore they exist $x$, such that $ |f(x)−f(y)|=2$. Then $ φf(δ)=2 $ for any $δ>0$.
how are sections b and c? I appreciate the help