I’m having trouble understanding the notion of uniform continuity, the definition states as follows:
Let $f: Dtomathbb{R}$, $f$ is uniformly continous in $Xsubset D$ if:
$$forallvarepsilon >0, existsdelta >0:left ( forall x_1, x_2in X, |x_1 – x_2|<deltaRightarrow |f(x_1)-f(x_2)|<varepsilonright ) $$
In other words, if a function is uniformly continous in $X$ then it is continous in $X$, but the contrary might not always be true.
As an example, consider $f(x) = x^2$. Fix $varepsilon =1$, then we can consider $$x_1 =frac{1}{delta}, x_2 = frac{1}{delta} + frac{delta}{2}Rightarrow |x_1-x_2 |= frac{delta}{2}<deltaRightarrow |x^2_1 – x^2_2| > 1 $$
But now Cantor’s theorem on uniform continuity states that if $f$ is continous in a closed interval $[a,b]$, then it is also uniformly continous in $[a,b]$.
Did we not just provide an example of a closed interval $[frac{1}{delta}, frac{1}{delta}+frac{delta}{2}]$ where uniform continuity is not satisfied?