Question: I cannot understand why the following statement is false:

"Let $f$ be a function defined as $f:R^2\to R$ such that $f(0,0)=1$. If for all $\varphi\in[0,2\pi[$ fixed we have $\lim\_{r\to 0}f(rcos(\varphi),rsin(\varphi))=1$ then $f$ is continuous at $(0,0)$."

To me it seems that this is exactly the same concept as taking a limit in polar coordinates. In this case we would normally see if the limit exists independently of $\varphi$. Isn't this what we are doing here?

Answer: 
**Hint/Sketch of proof:** The complementary language, i.e. language of strings which either do not start by "$010$" or are palindromes, can be shown not to be regular by using pumping lemma. It's roughly in the same proof you do when you show that the language of palindrome strings is not regular.

*The idea:* Pick $L$ said language, $n$ as in pumping lemma. Pick a string $w\in L$ such that:

1. $w$ is palindrome
2. $w$ starts (and therefore ends) with "$010$"
3. the length of $w$ is $\ge 10n$ (say)

So write $w=uvx$ as in pumping lemma. The idea is that, if $w=010z$, then $uv^ix=010z'$ as well, whatever the decomposition. Now, if you choose appropriately $w$ satisfying (1),(2),(3) you can find one such that $uv^ix$ cannot be palindrome.
HINT: For each $n\in\Bbb Z^+$ let $w\_n=01(01)^n1$: $w\_1=01011$, $w\_2=0101011$, $w\_3=010101011$, and so on. Similarly, let $z\_n=(01)^n0$. Show that $w\_mz\_n$ is a palindrome if and only if $m=n$, and apply the [Myhill-Nerode theorem](https://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem).