Proof/disprove contunuity of a map

Question

I need help with proving / disproving something:

Look at the map

$$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$

a) Is it continuous?
b) Is uniformly continuus?
c) Can one find a non-empty subset of $C([0,1],\mathbb R)$ in which $\Phi$ is Lipschitz-continuous?

Can you help me with this?

Answer 1

Hints:

For (a) & (c):

$\Phi(u)-\Phi(u') = \int (u-u')(u+u')$ and so $\|\Phi(u)-\Phi(u')\| \le \|u-u' \| \| u+u' \|$.

If you just consider constant functions you should be able to answer (b).