Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$

Question

$\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | ); \: \: \: \: \: \: \phi(u):=\int_0^1 u^2(t) dt $

Is this function continuous or even uniformly continuous? (I know that the function $g: M\to \Bbb R , g(x) := x^2$ is continuous but not uniformly)

Also, is there a non-empty subset of $ C([0,1];\Bbb R )$ so that $\phi$ is Lipschitz-continuous on that set?

Thanks in advance!

Answer 1

$\phi$ will be continuous. Indeed, fix any $u\in C_{[0,1],\mathbb{R}}$, and let $\delta > 0$ such that $\delta(2\lVert u\rVert_\infty + \delta) \leq \varepsilon$.

Fix any $\varepsilon > 0$. If $v$ is such that $\lVert u - v\rVert_\infty \leq \delta$, then $$\begin{align} \lvert \phi(u)-\phi(v) \rvert &= \left\lvert \int_{[0,1]} u^2-v^2\right\rvert \leq \int_{[0,1]} \left\lvert u^2-v^2\right\rvert \\ &= \int_{[0,1]} \underbrace{\left\lvert u-v\right\rvert}_{\leq \lVert u - v\rVert_\infty} \cdot \underbrace{\left\lvert u+v\right\rvert}_{\leq 2\lVert u\rVert_\infty+\delta} \leq \delta(2\lVert u\rVert_\infty + \delta) \leq \varepsilon \end{align}$$ i.e. $\phi$ is continuous.

It is easy to adapt the above to show that $\phi$ will be Lipschitz (and therefore uniformly continuous) on any subset of any $C_{[0,1],[-a,a]}\subseteq C_{[0,1],\mathbb{R}}$ (i.e., for functions bounded by some universal constant). The Lipschitz constant will then be $2a$ (replacing the $2\lVert u\rVert_\infty + \delta$ term).