diff --git a/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf index 3d69383..a24de26 100644 Binary files a/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf and b/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf differ diff --git a/semester3/analysis-ii/parts/vectors/differentiation/00_continuity.tex b/semester3/analysis-ii/parts/vectors/differentiation/00_continuity.tex index 98b4eb3..c2c177f 100644 --- a/semester3/analysis-ii/parts/vectors/differentiation/00_continuity.tex +++ b/semester3/analysis-ii/parts/vectors/differentiation/00_continuity.tex @@ -25,7 +25,16 @@ We write $\lim_{\elementstack{x \rightarrow x_0}{x \neq x_0}} f(x) = y$ iff $\forall (x_k)$ in $X$ s.t. $x_k \rightarrow x$ as $k \rightarrow +\infty$ and $x_k \neq x_0$ $(f(x_k))$ in $\R^m$ converges to $y$ \stepLabelNumber{all} \shortproposition Let $X \subseteq \R^n$, $y \subseteq \R^m$, $p \in \N$ and let $f: X \rightarrow Y$ and $g: Y \rightarrow \R^p$ be cont. Then $g \circ f$ is continuous +% ──────────────────────────────────────────────────────────────────── +\numberingOff +\inlineremark To find the limits, we have two tricks (for $\limit{(x, y)}{(a, b)}$): +\rmvspace +\begin{enumerate}[noitemsep] + \item \bi{(Substitution)} Substitute $y = x + (b - a)$, then limit is $\limit{x}{(a - b)}$ + \item \bi{(Polar coordinates)} Substitute $x = r \cos(\varphi)$ and $y = r \sin(\varphi)$ and the limit is $\limit{r}{0}$ +\end{enumerate} +\numberingOn\rmvspace \shortex \bi{(1)} $f_1 : \R^n \rightarrow \R^{m_1}$ and $f_2 : \R^n \rightarrow \R^{m_2}$ continuous $\Rightarrow f = (f_1, f_2): \R^n \rightarrow \R^{m_1 + m_2}$ is continuous (Cartesian product) \bi{(2)} Any linear map $f: \R^n \rightarrow \R^m$ is continuous. In particular, the identity map is continuous