I need help with #1 and #2. Thank you 1. **Problem:** Let \( f : \mathbb{R} \to \mathbb{R} \) be continuous. If \( f(c) > 0 \), prove there exists a neighborhood \( N \) of \( c \) such that \( f(N) > 0 \).2. **Problem:** Let \( f : \mathbb{R} \to \mathbb{R} \) and \( E \subseteq \mathbb{R} \). Prove \( f \) is continuous if and only if \( f^{-1}(E) \) is closed whenever \( E \subseteq \mathbb{R} \) is closed.3. **Problem:** Let \( f, g : D \to \mathbb{R} \) be continuous at \( c \in D \). Prove that \( fg \) is continuous at \( c \).4. **Definition:** A set \( X \subseteq \mathbb{R} \) is said to be disconnected if there exist disjoint open sets \( U \) and \( V \) such that \( X \subseteq U \cup V \), \( X \cap U \neq \emptyset \), and \( X \cap V \neq \emptyset \). A set is said to be connected if it is not disconnected. Let \( f : \mathbb{R} \to \mathbb{R} \) and \( X \subseteq \mathbb{R} \) be connected. Prove \( f(X) \) is connected.5. **Problem:** Suppose \( f : [a, b] \to [a, b] \) is continuous. Prove there exists \( c \in [a, b] \) such that \( f(c) = c \). (Hint: Consider the function \( g(x) = f(x) - x \).)

Name: I need help with #1 and #2. Thank you 1. **Problem:** Let \( f : \math
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: I need help with #1 and #2. Thank you 1. **Problem:** Let \( f : \mathbb{R} \to \mathbb{R} \) be continuous. If \( f(c) > 0 \), prove there exists a neighborhood \( N \) of \( c \) such that \( f(N) > 0 \).2. **Problem:** Let \( f : \mathbb{R} \to \m