The image contains a theorem and its proof from a mathematical text on continuous functions on the real numbers, $\mathbb{R}$.

Content Summary:

• Theorem 6.3: Let $f: I \to \mathbb{R}$ be any function, where $I \subseteq \mathbb{R}$ is an interval. Then $f$ is continuous on $I$ if and only if for any sequences $(x_n)$ and $(y_n)$ in $I$ that both converge to the same limit $c \in I$, the sequence $(f(x_n), f(y_n))$ converges to $(f(c), f(c))$.

• Proof Outline:

  • Let $I$ be any interval in $\mathbb{R}$, and $f: I \to \mathbb{R}$ a function.
  • Assume that $f$ is continuous on $I$.
  • To prove the desired statement, it is sufficient to show that for any sequences $(x_n) \to c$ and $(y_n) \to c$, we have $(f(x_n), f(y_n)) \to (f(c), f(c))$.
  • Continuity ensures that $f(x_n) \to f(c)$ and $f(y_n) \to f(c)$ as $n \to \infty$.

Would you like me to provide a more detailed explanation of the proof or clarify a specific part of it? Here are some related questions:

1. What is the definition of a continuous function?
2. How does sequence convergence relate to function continuity?
3. Why is the condition involving sequences $(x_n)$ and $(y_n)$ necessary for continuity?
4. Can you provide an example of a function that is not continuous and how it fails this theorem's condition?
5. How does this theorem relate to the epsilon-delta definition of continuity?

Tip: Remember that understanding the behavior of functions through sequences can provide deeper insight into their continuity properties!