To prove that $f(x) = x^2$ is continuous on $\mathbb{R}$, we will use the ε-δ definition of continuity. According to this definition, a function $f(x)$ is continuous at a point $x = a$ if, for every $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x - a| < \delta$, it follows that $|f(x) - f(a)| < \epsilon$.

We need to show that for any $a \in \mathbb{R}$, the function $f(x) = x^2$ satisfies this definition.

1. Let $f(x) = x^2$ and choose any point $a \in \mathbb{R}$.

2. We want to show that for every $\epsilon > 0$, there exists a $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - f(a)| < \epsilon$.

3. Compute $|f(x) - f(a)|$: $|f(x) - f(a)| = |x^2 - a^2|$ We can factor this expression as: $|x^2 - a^2| = |(x - a)(x + a)|$ Thus, $|f(x) - f(a)| = |x - a| \cdot |x + a|$

4. Control $|x + a|$: To ensure that $|f(x) - f(a)| < \epsilon$, we need to find an appropriate $\delta$ in terms of $\epsilon$. Notice that if we can bound $|x + a|$, we can find a relationship between $|x - a|$ and $\epsilon$.

   Assume that $|x - a| < 1$, which implies $x$ is in the interval $(a - 1, a + 1)$. Then: $|x + a| \leq |a + 1| + |a| = 2|a| + 1$ So, we have the bound $|x + a| \leq 2|a| + 1$ when $|x - a| < 1$.

5. Choose $\delta$: Now, to ensure $|f(x) - f(a)| < \epsilon$, we need: $|x - a| \cdot |x + a| < \epsilon$ Since $|x + a| \leq 2|a| + 1$ when $|x - a| < 1$, it suffices to choose $\delta$ such that: $\delta (2|a| + 1) < \epsilon$ Solving for $\delta$, we get: $\delta = \frac{\epsilon}{2|a| + 1}$

6. Conclusion: Given any $\epsilon > 0$, choose $\delta = \min \left(1, \frac{\epsilon}{2|a| + 1}\right)$. Then, for $|x - a| < \delta$, we have $|f(x) - f(a)| < \epsilon$. Therefore, $f(x) = x^2$ is continuous at $x = a$.

Since $a$ was arbitrary, $f(x) = x^2$ is continuous on $\mathbb{R}$.

Would you like further details on any part of the proof?

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Here are some related questions to expand on this topic:

1. How does the continuity of polynomial functions extend to all of $\mathbb{R}$?
2. What would be the proof for the continuity of $f(x) = x^3$ on $\mathbb{R}$?
3. How does the ε-δ definition of continuity compare with limits-based definitions?
4. Can the continuity of $f(x) = x^2$ be generalized to functions of higher degrees?
5. How would you prove continuity of $f(x) = x^2$ using sequences?

Tip: For polynomial functions, continuity can often be more easily shown by leveraging that they are compositions and sums of continuous functions, all continuous over $\mathbb{R}$.