Category: 機能の継続性

Definition of Continuous Function

Definition. A function $$$f$$$ is continuous at $$${a}$$$ if $$$\lim_{{{x}\to{a}}}={f{{\left({a}\right)}}}$$$.

Continuity implies three things:

  1. $$${f{{\left({a}\right)}}}$$$ is defined (i.e. $$${a}$$$ is in the domain of $$${f{}}$$$);
  2. $$$\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}$$$ exists;
  3. $$$\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}={f{{\left({a}\right)}}}$$$.

Geometrically, continuity means that you can draw a function without taking your pen off the paper.

One-Sided Continuity. Classification of Discontinuities

Similarly to the one-sided limits, we can define one-sided continuity.

Definition. Function $$${f{{\left({x}\right)}}}$$$ is continuous from the right at point $$${a}$$$ if $$$\lim_{{{x}\to{{a}}^{+}}}={f{{\left({a}\right)}}}$$$. Function $$${f{{\left({x}\right)}}}$$$ is continuous from the left at point $$${a}$$$ if $$$\lim_{{{x}\to{{a}}^{{-}}}}{f{{\left({x}\right)}}}={f{{\left({a}\right)}}}$$$.

Theorems involving Continuous Functions

Intermediate Value Theorem. Suppose that $$$f$$$ is continuous on closed interval $$${\left[{a},{b}\right]}$$$ and let $$${N}$$$ is any number between $$${f{{\left({a}\right)}}}$$$ and $$${f{{\left({b}\right)}}}$$$ (or $$${f{{\left({b}\right)}}}$$$ and $$${f{{\left({a}\right)}}}$$$; depends what is bigger). Then there exists number $$${c}$$$ in $$${\left({a},{b}\right)}$$$ such that $$${f{{\left({c}\right)}}}={N}$$$.