## Appendix A Proofs of Lemmas 1 and 2
Lemma A2: _There exists a computable sequence \((f_{M})_{M\in\mathbb{N}}\) of elementary computable functions in \(\mathcal{CB}^{1}_{\pi}\) such that \(\|f_{M}\|_{1}\geq 1\), \(\|\mathcal{S}_{1}f_{M}\|_{1}<\
icefrac{{1}}{{M}}\), and \(\|f_{M}\|_{1}\leq C\) are satisfied for all \(M\in\mathbb{N}\) and some number \(C>0\)._  
Recall that Lemma A1 implies the discontinuity of \(\mathbf{T}_{\infty}\), as follows from (A1). In the same manner, Lemma A2 implies the discontinuity of \(\mathbf{T}_{1}\): For \((f_{n})_{n\in\mathbb{N}}\) as in Lemma A2, we have  
\[\lim_{n\to\infty}\frac{\|\mathbf{T}_{1}\mathcal{S}_{1}f_{n}\|_{1}}{\| \mathbf{S}_{1}f_{n}\|_{1}}=\lim_{n\to\infty}\frac{\|f_{n}\|_{1}}{\|\mathbf{S}_{1}f_{n }\|_{1}}>\lim_{n\to\infty}\frac{1}{1/n}=\infty\] (A4)  
Proof of Lemma A2. : For all \(M\in\mathbb{N}\) and with \(k(m):=-2m\) for \(m\in\mathbb{N}\), define the function \(f^{\prime}_{M}\) according to  
\[f^{\prime}_{M}:=\operatorname{sinc}-\frac{1}{M}\sum_{m=1}^{M} \operatorname{sinc}_{k(m)}.\]  
First, observe that \(\|f^{\prime}_{M}\|_{1}<4+(\
icefrac{{5}}{{\pi}})\ln(2M+1)\) is satisfied for all \(M\in\mathbb{N}\). For a detailed proof, we refer directly to Equation (18) in [19], where the relevant inequality is explicitly derived. Since in addition, \(\dot{f}^{\prime}_{M}\) is a finite linear combination of sinc-functions, we consequently have \(f^{\prime}_{M}\in\mathcal{CB}^{1}_{\pi}\) for all \(M\in\mathbb{N}\). Second, observe that for all \(M\in\mathbb{N}\), we have \((\
icefrac{{1}}{{6\pi}})\ln(\
icefrac{{M}}{{2}})-(\
icefrac{{1}}{{\pi}})<\|f^{ \prime}_{M}\|_{1}\), which we can deduce from Equation (11) in [19] with a few additional steps. We define \(\eta:\mathbb{R}\to\mathbb{R}\), \(t\mapsto\eta(t)\),  
\[\eta(t):= \begin{cases}0&\text{if }t<0,\\ t\sin(\pi t)&\text{if }0\leq t<1,\\ \sin(\pi t)&\text{otherwise}.\end{cases} \]  
Then, using Equation (11) in [19], we obtain the chain of inequalities  
\[\frac{1}{6\pi}\ln\left(\frac{M}{2}\right)-\frac{1}{\pi}<\int_{0 }^{\infty}f^{\prime}_{M}(t)\eta(t)\;\mathrm{d}t\overset{\text{(a)}}{<}\int_{- \infty}^{\infty}|f^{\prime}_{M}(t)||\eta(t)|\;\mathrm{d}t\overset{\text{(b)}}{ \leq}\int_{-\infty}^{\infty}|f^{\prime}_{M}(t)|\;\mathrm{d}t=\|f^{\prime}_{M}\| _{1}\]  
for the function \(f^{\prime}_{M}\) and all \(M\in\mathbb{N}\), where (a) follows from the fact that \(a(t)=0\) for all \(t<0\) and (b) follows from the fact that \(|a(t)|\leq 1\) for all \(t\in\mathbb{R}\). For \(m\in\Omega\), there exists \(k\in\mathbb{N}\) such that for all \(l\in\mathbb{N}\) that satisfy \(l\geq k\), we have \(f_{m,l}=f_{m,k}\), i.e., the limit value \(\lim_{l\to\infty}f_{m,l}\) exists and is an elementary computable function. Furthermore, there exists an \(M\in\mathbb{N}\) such that \(f_{M}=\lim_{l\to\infty}f_{m,l}\) is satisfied. We define  
\[f_{m}^{\prime}:= \begin{cases}\lim_{l\to\infty}f_{m,l}&\text{if }m\in\Omega,\\ 0&\text{otherwise},\end{cases} \]  
for all \(m\in\mathbb{N}\). Hence, \((f_{m}^{\prime})_{m\in\mathbb{N}}\) is a (_not_ necessarily computable) sequence of elementary computable functions. Furthermore, the sequence \((x_{m}^{\prime})_{m\in\mathbb{N}}:=(\mathbb{S}_{1}f_{m}^{\prime})_{m\in \mathbb{N}}\) is a sequence of elementary computable sequences. In the following, we will prove by case distinction that for all \(m\in\mathbb{N}\), the sequence \((x_{m,k})_{k\in\mathbb{N}}\) converges effectively towards \(x_{m}^{\prime}\) in \(\ell_{0}^{1}\), with respect to the \(\mu\)-recursive modulus of convergence \(\xi:\mathbb{N}^{2}\to\mathbb{N},\ (m,K)\mapsto K\). First, assume that \(m\in\Omega\) is satisfied. Then, there exists \(k\in\mathbb{N}\) such that for all \(K\in\mathbb{N}\) that satisfy \(K\geq k\), we have \(x_{m,K}=x_{m,k}\). Consider the smallest such \(k\in\mathbb{N}\) and observe the following for all \(K\in\mathbb{N}\):  
* If \(K\in\mathbb{N}\) satisfies \(K\geq k\), we have \(\|x_{m}^{\prime}-x_{m,K}\|_{1}=\|x_{m,k}-x_{m,k}\|_{1}=0<2^{-K}\).