The following function series is given.
For $(f_n)_{n \in \mathbb{N}}$ in $C^0([-1,1])$
\begin{equation}
f_n(x) =
\begin{cases}
0 & -1 \le x \le -\frac{1}{n} \\
1 +nx & -\frac{1}{n} < x \le 0 \\
1 & 0 < x \le 1
\end{cases}
\end{equation}
What does the symbol $C^0([-1,1])$ mean?
Sorry if this question is very basic, but googling didn't help.
Best Answer
For any set $X$, $C^0(X)$ is simply the set of all real-valued (or complex-valued) continuous functions on the set $X$. It is also often denoted $C(X)$.
Generally, $C^k(X)$ denotes the set of functions on $X$ for which at least the first $k$ (partial) derivatives exist and are continuous. So with $k=0$, there are no differentiability requirements, and $C^0$ contains all the continuous functions, differentiable or not.