Continuity of maps between function spaces and normed spaces

Will the mapping $f$ of the space $C[0 ; 1]$ (see problem 31) into itself be continuous, if it is given by the formula:

Will the mapping $f(x) = x^{2}$ of the space $C L_{1}[0 ; 1]$ (see problem 3.5) into itself be continuous?

Let $X = \left\{ x \in C^{1}[0 ; 1], x(0) = 0\right\}$ (see problem 93). Will the mapping $f: X \rightarrow C[0 ; 1]$, given by the formulas $f(x)(t) = \frac{x(t)}{t}, t \in (0 ; 1], f(x)(0) = \lim_{t \rightarrow 0} \frac{x(t)}{t}$, be continuous?

Are the following mappings of the space $C[0 ; 1]$ (see problem 31) into itself continuous:

Are the following mappings $f$ of the space $C L_{2}[0 ; 1]$ (see problem 3, part 8) into itself continuous:

Will the mapping $f(x) = x(1)$ be continuous, if it is considered as a mapping:

Is differentiation, regarded as a mapping of the subspace consisting of continuously differentiable functions of the space $C[a ; b]$ (see problem 31) into the space $C[a ; b]$ itself, continuous?

Is differentiation, regarded as a mapping of the space $C^{1}[a ; b]$ (see problem 93) onto the space $C[a ; b]$ (see problem 31), continuous?

Let $F: C[0,1] \rightarrow \mathbb {R}$ be defined by $F(f)=f(0)$, and let $g(x)=x^{2}$. Find $\lim_{f \rightarrow g} F(f)$.

Let $d$ be a metric on $C[0,1]$ defined by

(see Exercise 2.1.3). Let $F: C[0,1] \rightarrow \mathbb {R}$ be defined by $F(f)=f(0)$, and let $g(x)=x^{2}$. Find $\lim_{f \rightarrow g} F(f)$.

Let $d$ be a metric on $\mathbb {R}$ defined by

(see Exercise 2.1.8), and let $\rho$ denote the usual distance on $\mathbb {R}, \rho (x, y)=\left|x-y\right|$. Let $f:(\mathbb {R}, d) \rightarrow (\mathbb {R}, \rho )$ be defined by $f(x)=x^{2}$. Find $\lim_{x \rightarrow 2} f(x)$.

For a normed linear space $X$, show that the function $\left\| \cdot \right\| : X \rightarrow \mathbb {R}$ is continuous.

Let $X$ be a normed linear space.

Verify that the functions $T_{i}, i = 1,2,3$ defined by (4.42) are continuous on $\mathcal{C}[0,1]$.