Non uniform continuity of the function $\sin\left( \frac{1}{2+\sin(t)+\sin(t\sqrt2)}\right)$

Question

How can we show that the function $$f(t)=\sin\left( \frac{1}{2+\sin(t)+\sin(t\sqrt2)}\right)$$ is not uniformly continuous on $\mathbb{R}$.

This question gives a sequence $(t_n)_n$ such that $$\frac{1}{2+\sin(t_n)+\sin(t_n\sqrt2)}\to+\infty$$ when $n\to+\infty$.

Answer 1

A function $f(t)$ over $\mathbb R$ is uniformly continuous iff $\forall \varepsilon > 0, \exists \delta > 0$ such that $\forall x, y \in X, |x − y| < \delta \implies |f(x) − f(y)| < \varepsilon$.

Note that this is equivalent to saying that $f$ is uniformly continuous iff $\forall \varepsilon > 0, \exists \delta > 0$ such that $\forall x, y \in X, |x - y| \geq \delta \vee |f(x) - f(y)| < \varepsilon$.

Now, if we negate this, we see that $f$ is not uniformly continuous iff $\exists \varepsilon > 0$ such that $ \forall \delta > 0, \exists x, y \in X$ such that $ |x - y| < \delta \wedge |f(x) - f(y)| \geq \varepsilon$.

All that remains to show that $f$ is not uniformly continuous is to find $\varepsilon, x(\varepsilon),$ and $y(\varepsilon)$ that satisfy this statement.