the addition,subraction and multiplication operations are continuous functions from RxR into R and the quotient operation is a continuous function from Rx(R-{0})into R ,prove using metric topology

To prove that the addition, subtraction, and multiplication operations are continuous functions from \( \mathbb{R} \times \mathbb{R} \) into \( \mathbb{R} \), and that the quotient operation is a continuous function from \( \mathbb{R} \times (\mathbb{R} \setminus \{0\}) \) into \( \mathbb{R} \) using metric topology, we can proceed as follows:

### Metric Topology Recap:
In metric topology, a function \( f: X \to Y \) between two metric spaces \((X, d_X)\) and \((Y, d_Y)\) is continuous if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( d_X(x_1, x_2) < \delta \) in \( X \), it follows that \( d_Y(f(x_1), f(x_2)) < \epsilon \) in \( Y \).

### Addition and Subtraction:
Let \( f: \math

Explore the proof of continuity for addition, subtraction, multiplication, and division operations on real numbers using metric topology. This mathematical analysis demonstrates how these operations maintain continuity under varying conditions.

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