Show that the following functionals are differentiable in following spaces:
(1): $J\left[y\right] = y(a)\ \ \ in \ \ \ C\left[a,b\right];$
(2): $J\left[y\right] = y(a)\ \ \ in \ \ \ C^{'}\left[a,b\right];$
(3): $J\left[y\right] = \sqrt{1+y^{'2}(a)}\ \ \ in \ \ \ C^{'}\left[a,b\right];$
(4): $J\left[y\right] = \left|y(a)\right|\ \ \ in \ \ \ C\left[a,b\right].$
For (1) and (2) it's clear: $\Delta J=J\left[y+\delta y\right]-J\left[y\right]=\delta y$. It's differentiable, because $\Delta J$ equals to Linear functional of $\delta y$. But, for (3) and (4) it's not very clear. If for (4) I can write: $\Delta J=J\left[y+\delta y\right]-J\left[y\right]=\left|y(a)+\delta y\right|-\left|y(a)\right|$ - here we can see that the first term is a linear functional of $\delta y$ , but second term does not goes to 0, thus it's not differentiable. Book says (3) is differentiable, I have no idea how to show it.