Examples for when differentiability fails

Question

Let $l^1(\mathbb{N};\mathbb{R})$ be the set of all sequences $\mathbb{N}\to\mathbb{R}$ such that $\sum_{n\in\mathbb{N}}|x_n|<\infty$ for all $x\in l^1(\mathbb{N};\mathbb{R})$, together with the norm $||x||_{l^1(\mathbb{N};\mathbb{R})}=\sum_{n\in\mathbb{N}}|x_n|$.

1. What is an example of a map $f:l^1(\mathbb{N};\mathbb{R})\to\mathbb{R}$ which is not Frechet differentiable at some point $y\in l^1(\mathbb{N};\mathbb{R})$, even though there is a linear map $A:l^1(\mathbb{N};\mathbb{R})\to\mathbb{R}$ such that $$ \lim_{h\to0}\frac{|f(y+h)-f(y)-A(h)|}{||h||_{l^1(\mathbb{N};\mathbb{R})}} =0 $$ but this map $A$ is not continuous and so $f$ is not Frechet differentiable at $y$?

2. What is an example of a map $f:l^1(\mathbb{N};\mathbb{R})\to\mathbb{R}$ which is not Frechet differentiable at some point $y\in l^1(\mathbb{N};\mathbb{R})$, even though there is a continuous map $A:l^1(\mathbb{N};\mathbb{R})\to\mathbb{R}$ such that $$ \lim_{h\to0}\frac{|f(y+h)-f(y)-A(h)|}{||h||_{l^1(\mathbb{N};\mathbb{R})}} =0 $$ but this map $A$ is not linear and so $f$ is not Frechet differentiable at $y$?

EDIT

For question 1, is it possible to find such an example where $f$ itself is continuous?