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Let's break down your request step by step: --- ## **Step 1: Definition of a Regular Function** A **regular function** (in calculus, particularly in the context of real analysis and algebra) generally means a function that is **continuous and differentiable** in its domain. However, the term **"regular function"** can have different meanings based on context: - In **complex analysis**, a regular function is another name for a **holomorphic** (or analytic) function—one that is complex differentiable in a neighborhood of every point in its domain. - In **real analysis**, a regular function can mean a function that is **continuous** (and often also differentiable). - In **algebraic geometry**, a regular function on an affine variety is a polynomial function. In calculus or basic mathematics, when someone asks for a "regular function," they typically mean a function that is **continuous and differentiable everywhere on its domain**. --- ## **Step 2: Example of a Regular Function** Let's take the function: \[ f(x) = x^2 \] ### **Why is \( f(x) = x^2 \) a regular function?** - It is **defined** for all real numbers (\( x \in \mathbb{R} \)). - It is **continuous** everywhere. - It is **differentiable** everywhere (the derivative \( f'(x) = 2x \) exists for all \( x \)). --- ## **Step 3: Step-by-Step Check** 1. **Is it continuous?** - Yes, \( x^2 \) is a polynomial, and all polynomials are continuous everywhere. 2. **Is it differentiable?** - Yes, the derivative exists for all \( x \). 3. **Is it defined for all real numbers?** - Yes. --- ## **Final Answer** ### **Definition:** A **regular function** is a function that is continuous and differentiable everywhere in its domain. ### **Example:** \[ f(x) = x^2 \] This function is continuous and differentiable for all real values of \( x \), so it is a regular function. --- **Summary Table:** | Function | Continuous? | Differentiable? | Regular? | |---------------|-------------|-----------------|----------| | \( x^2 \) | Yes | Yes | Yes | | \( |x| \) | Yes | No at \( x=0 \) | No | | \( \sin(x) \) | Yes | Yes | Yes | --- **In conclusion:** A regular function is continuous and differentiable everywhere. **Example:** \( f(x) = x^2 \).