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Math and science::Analysis::Tao::09. Continuous functions on R

Equivalent formulations of function continuity

Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( x_0 \) be an element of \( X \). Then the following four statements are logically equivalent:

  1. \( f \) is continuous at \( x_0 \).
  2. For every sequence \( (a_n)_{n=0}^{\infty} \) consisting of elements of \( X \) which converges to \( x_0 \), the sequence \( (f(a_n))_{n=0}^{\infty} \) converges to \( f(x_0) \).
  3. [...]
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