# Sergei Yakovenko's blog: on Math and Teaching

## Monday, November 30, 2009

### אוסף בעיות מספר 3

לפוסט זה מצורף אוסף בעיות מספר 3. אם עולות שאלות בקשר לבעיות לפני מפגש התרגול הבא, אפשר לשאול כאן.

## Continuity

A function $f\colon X\to\mathbb R$ defined on a subset $X\subseteq\mathbb R$ is continuous (in full, continuous on $X$), if it is continuous at each point $a\in X$. Continuity is equivalent to the requirement that the preimage $f^{-1}(U)$ of any open set $U$ is open in $X$ (i.e., is an intersection between $X$ and an open subspace $V\subseteq \mathbb R$).

• Continuity at a point is a local property. Automatic continuity at isolated points of $X$.
• Examples of discontinuity points.
• Continuity and arithmetic operations. Continuity of the composition. Continuity of elementary functions.
• Dirichlet function: the ugly beast. Further pathologies.
• Continuity and global properties:
1. Continuity and boundedness
2. Continuity and existence of roots
3. Continuity and monotonicity
• Continuity and functional equations $f(x+y)=f(x)+f(y)$, $f(x+y)=f(x)\cdot f(y)$, $f(xy)=f(x)+f(y)$, $f(xy)=f(x)\cdot f(y)$.

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