Sergei Yakovenko's blog: on Math and Teaching

Monday, November 30, 2009

Lecture 5 (Dec 1)

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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