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Narayan lets us in on the Tamil novel known as “Nandhipurathu nayagi” — and yes, it is, he tells us, an example of.Q:

On the continuity of the Weierstrass Approximation theorem

Is it true that if $f$ is continuous on $[0,1]$ and has a convergent Taylor expansion at $x=0$, then $f$ must be continuous on $\mathbb{R}$?


Continuity of $f$ at $x=0$ implies continuity on $[0,1]$ by uniform continuity (since $f$ can be written as a composition of $g(x)=x$ and $h(x)=\frac{f(x)}{x}$).
Convergence of the Taylor expansion at $x=0$ means that $f(x)=x+O(x^2)$ with $O(x^2)$ uniform in $x$. If you choose $x=\frac{1}{n}$ the previous bound implies that $x+O(x^2) \geqslant \frac{1}{n}+o(1)$, whence $f(x) \geqslant \frac{1}{n}+o(1)$. But the lower bound is also valid for $\frac{1}{n+1}$, $n \geqslant 1$. This is a contradiction.
Therefore, the Taylor expansion must diverge at $x=0$.
Edit. If $|f(x)| \leqslant |x|+\varepsilon$ on $[0,1]$, then the upper bound is valid on $\mathbb{R}$. The idea of the proof is the same as above.

n$th-order $K$-theory of $R$, which is a sort of completion of the $K$-theory of $R$ at the $n$th level.

[^4]: This can be shown directly by considering the tensor product of the statements that $\operatorname{Tor}^P_{p,q}(A

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Thiilanjli is a successfull Tamil novel written by Pazhani Janakarajan popularly known as Pazhani. Tamil Novels(Tamil:  :  :   ) are popular in Tamil culture and are considered a great source of Tamil Literature.Tamil Novels (Tamil:  :  :   ), or as they are also called, Tamil novels (Tamil:  :  :   ), are the written works written in the Tamil language.
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He is a contemporary artist and they write a number of books about him. His first novel was published in Hindi. His first novel which was written in Hindi was published
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Nandhipurathu Nayagi Novel is a famous novel of Pranayanan Umachandran. Pranayanan Umachandran and his wife Sumathi have written a number of books about their family.
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