I thank Willie Wong for giving such a nice answer which indeed cleared some of my misconceptions. I am giving my views on the question especially considering the context in which this question is being asked. I also give some comments on the question mentioned in the motivation part and also the answer given there. I am posting this in the space for answer due to absence of a better place. Please let me know if there is a better alternative. Also request you to comment upon any mistakes in this post.
Outline
We try to give a more general view of the problem and argue that it is important especially in the context (given in the motivation part of the question) in which the question is being asked. In the motivation part of the question a problem was cited as a link and here we refer to it as Problem A. We also discuss some possible approaches to Problem A.
In Section 1 we give some definitions based on continuity and differentiability of a function.In Section 2 we give two theorems which are fundamental tools in the current context. In Section 3 we consider a norm denoted as $\|\cdot\|$ defined on the linear space $X = C^r([0,1])$ and explain how it is tacitly designed to make the normed linear space $(X, \|\cdot\|)$ complete. We also attempt to give a proof for this using the theorems given in section 2. Section 4 is about convergence of functions. In Section 5 we give an example of a normed linear space whose norm is of the form $\|\cdot\|$ which need not necessarily be complete. We also point out some limitations of convergence under such a norm. In Section 6 we give some closing remarks.
Section 1
Here we give some definition based on continuity and differentiability of a function.
Continuity in an interval :
The following two statements are equivalent.
- A function $f$ is continuous in an interval $S$.
- A function $f$ is continuous at every point $x \in S$.
Differentiability in an interval :
The following two statements are equivalent.
- A function $f$ is differentiable in an interval $S$.
- A function is differentiable at every point $x \in S$.
$\mathcal{C}^k$ in an interval :
A function which is atleast $k$ times continuously differentiable at every point $x \in S$ is said to be $\mathcal{C}^k$ in $S$.
The collection of all functions $f$, such that $f$ has support in $S$ and is atleast $k$ times continuously differentiable at every point $x \in S$, is denoted as $\mathcal{C}^k(S)$.
Degree of differentiability :
- The degree of differentiability of a function $f$ at a point $x$ is $k$. $k \in \{0\}\cup\mathbb{N}\cup\{\infty\}$.
- A function $f$ is continuous at a point $x$ and the maximum number of times $f$ is continuously differentiable at $x$ is $k$.
Differentiability map :
Let a function $f : D \to R$ where $D\subseteq\mathbb{R}$ and $R\subseteq\mathbb{R}$. Let $C_f$ be the set of all points $x \in D$ such that $f$ is continuous at $x$.
The differentiability map of the function $f$ is defined as the mapping $dm_f : C_f \to \{0\}\cup\mathbb{N}\cup\{\infty\}$ such that $dm_f(x) = k$, where $k$ is the degree of differentiability of $f$ at the point $x$.
Hence associated with every function $f : \mathbb{R} \to \mathbb{R}$ there is a differentiability map $dm_f : C_f \to \{0\}\cup\mathbb{N}\cup\{\infty\}$ with $C_f$ defined as above.
Differentiability class :
The collection of all functions $f : \mathbb{R} \to \mathbb{R}$ which have $dm : \mathbb{R} \to \{0\}\cup\mathbb{N}\cup\{\infty\}$ as the differentiability map is called the differentiability class $D_{dm}$ of the differentiability map $dm$.
Note that there are differentiability maps whose differentiability classes are empty sets.
Section 2
Two theorems on uniform convergence to continuity and to differentiability are given here
Theorem 1:
Let $\{f_n\}$ be a sequence of functions which are continuous in an interval $S$. If the sequence of functions converges uniformly to $f$ in an interval $S$, then the limit function $f$ is continuous in the interval $S$.
Theorem 2:
Let $\{f_n\}$ be a sequence of functions which are differentiable in an interval $S$. If the sequence converges uniformly to a function $f$ in the interval $S$, and the sequence $\{f^'_n\}$, consisting of derivatives of $\{f_n\}$ in the interval $S$, also converges uniformly to a function $g$ in the interval $S$, then $f$ is differentiable in the interval $S$ and its derivative is $g$.
These theorems can be found here.
A slightly different version of Theorem 2 which tells about the differentiability of the limit function of a sequence of functions, at a specific point, is given below
Theorem 2A:
Let $\{f_n\}$ be a sequence of continuous functions defined in an interval $S$ and the sequence converges uniformly to a function $f$ in the interval $S$. Consider a point $x \in S$ and if there exists a neighbourhood $N$ of $x$ in $S$ such that the functions f_n $n = 1,2,3...$ are all differentiable in $N$ with derivatives $f^'_n$ defined in the interval $N$, and the sequence $\{f^'_n\}$ converges uniformly to a function $g$ in the interval $N$, then the function $f$ is differentiable at $x$ and its derivative at $x$ is given by $g(x)$.
Section 3
We now consider the linear space $X = C^r([0,1])$ with a norm $\|f\| = \sum_{i = 0}^r \sup_{x\in [0,1]} \left|\frac{d^i}{dx^i} f(x)\right|$ denoted as $\|\cdot\|$ defined over it. we now show how the norm $\|\cdot\|$ defined over $X$ is tacitly designed by keeping in mind the theorems given in Section 2, to make the normed linear space $(X, \|\cdot\|)$ complete. We do this by giving a proof for the completeness of $(X, \|\cdot\|)$ using the theorems in Section 2.
Consider the case r=0.
If any sequence $\{f_n\}$ where $f_n \in X$ is a Cauchy sequence under the given norm then it can be easily seen that it converges uniformly (note here that it is point-wise) to some function lets say $l$. From Theorem 1 and given that $f_n \in X$ it can be readily seen that the limit function $l \in X$.
Consider the case r > 0.
Consider a Cauchy sequence $\{f_n\}$, $f_n \to f$, $f_n \in C^r([0,1])$ and hence (from what followed earlier) $f \in C([0,1])$. The norm $\|\cdot\|$ is tacitly designed such that the sequences $\{f_n^k\}$, for $ k = 1,2,...r$ are all Cauchy under the given norm.Also $f_n^k \in C([0,1])$ for all $k = 1,2,...r$. Consider the sequence $\{f_n^'\}$ Now we can see from what has been proved earlier for the case r = 0, that the sequence $\{f_n^'\}$ converges uniformly to some function $m \in C([0,1])$. Now using Theorem 2 it readily follows that $f \in C^1([0,1])$. It can be proved that $f \in C^r([0,1]$ by using the Theorem 2 inductively and the fact that the sequences $\{f_n^k\}$, for $ k = 1,2,...r}$ are all Cauchy under the given norm.
Hence the normed linear space $(X, \|\cdot\|)$ is complete. The theorems mentioned in Section 2 play a crucial role in proving this.
Section 4
Consider the functions of the form $f : \mathbb{R} \to \mathbb{R}$, in order to define the convergence of a sequence of such functions, there are two ways of doing so, which are mentioned below.
Point-wise convergence : In order to define point-wise convergence we use the fact that $\mathbb{R}$ forms a metric space under the metric $d(x,y) = |x-y|$ where $x,y \in \mathbb{R}$.See the definitions of Point-wise convergence and Uniform convergence.
Convergence under a norm : To define the convergence of a sequence of functions under a norm, we need a space $X$ whose elements are functions and which contains all the functions in the given sequence, with a norm defined on it.
A sequence $\{f_n\}$ where $f_n \in X$ is said to be converging sequence If there exists a function $f \in X $ also called and the limit of the sequence, such that $\lim_{n\to \infty} \|f_n - f\| = 0$. This is also explained in the previous answer by Willie Wong.
Section 5
We now give an example of a linear space which when equipped with a norm of the form $\|\cdot \|$ need not necessarily be complete.
Let $dm : (0,1) \to \{0\}\cup\mathbb{N}\cup\{\infty\}$ be a differentiability map and $D_{dm}$ be its differentiability class such that $D_{dm}$ is nonempty.
Let $k = \min\limits_{x} dm(x)$. Let $X = \mathcal{C}^k((0,1))$. Therefore the linear space $D_{dm} \subset X$. Now we define a norm denoted by $\|\cdot \|$ given as $\|f\| = \sum_{i = 0}^k \sup_{x\in [0,1]} \left|\frac{d^i}{dx^i} f(x)\right|$, on the spaces $X$ to form two normed linear spaces $(X, \|\cdot\|)$ and $(D_{dm}, \|\cdot\|)$.
We know that $(X, \|\cdot\|)$ is complete. We however note that we note that $(D_{dm}, \|\cdot\|)$ need not necessarily be complete. We can also easily see that the approach of constructing a function $f \in D_{dm}$ as a limit function of a sequence of functions $\{f_n\}$, where $f_n \in X$, converging under the norm $\|\cdot\|)$ may not be such a good idea, as such a convergence only guarantees that the limit function belongs to $X$ but cannot guarantee for the limit function to have a desired differentiability map and hence cannot guarantee that the limit function belongs to $D_{dm}$.
On the other hand to construct a function $f \in D_{dm}$, the approach of constructing $f$ as a limit of sequence of functions $\{f_n\}$ converging under pointwise convergence seems to be a promising idea because the Theorem 2A, given in section 2, gives the sufficient conditions on the functions $f_n$ for the limit function to have a desired degree of differentiability at specified points in its domain.
Section 6
Here we give some closing remarks and also comment upon the answer given to Problem A and why such an approach is not a good idea.
In the answer the function $f$ is given as $f=\sum_{j < q} f_j + \sum_{j\geq q} f_j$. It is said that the first sum does what is needed at the points $p/q$, and the series converges in $C^q_c(\mathbb{R})$. Let us denote the first term as $f_1$ and let the series convergence to some $f_2 \in C^{\infty}_c(\mathbb{R}) \subset C^q_c(\mathbb{R})$. Consider a rational point $x_o = P/Q$ where $P,Q \in \mathbb{N}$ and $P \le Q$ and $P/Q$ in simplest form and $Q \ge q$. We can easily see that $Cf_1(x_o) = \infty$ and $Cf_2(x_o) = \infty$ and hence $Cf(x_o) = \infty$ which violates condition 2 (EDIT: its condition 1 instead of condition 2) of the problem. EDIT 2 : Please see a clarification to answer to Problem A given there.Now it is clear that the answer given there is true. But still this discussion (except for these comments on the answer) is valid in a more general case.
The approach of convergence of a sequence of functions under the norm of the form $\|\cdot \|$ given above is not suitable to get a limit function $f$ with a desired differentiability map.
On the other hand the approach of pointwise convergence is more fundamental and Theorem 2A gives the sufficient conditions on the functions in the sequence, to get a desired degree of differentiability for the limit function at specified points in its domain.