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

Function convergence's equivalence to sequence convergence

Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real. Then the following two statements are logically equivalent:

  • \( f \) converges to \( L \) at \( x_0 \) in \( E \).
  • [...].