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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

taniadiaz
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  • For (b) , it is $2$, when $x$ is a multiple of $\sqrt{(2n+1)\frac{\pi}{2}}$, and for (c) it is $\infty$ – PNDas Oct 26 '20 at 04:19
  • in parking lot c, how is it? It's infinite? or less infinity? I can't appreciate it. thanks for answering. – taniadiaz Oct 26 '20 at 10:38

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