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:
\( f \) is continuous at \( x_0 \).
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 \( f(x_0) \).
