Overview

This skill guides you through solving continuity problems in real analysis using a practical decision tree. It emphasizes verifying that f is defined at a and that lim x->a f(x) exists and equals f(a), with support from symbolic tools like SymPy and formal proofs via Z3.

How This Skill Works

The workflow starts by checking the definition of continuity at a: f(a) must be defined, the limit lim x->a f(x) must exist, and this limit must equal f(a). It then uses SymPy to compute limits and compare them to f(a); for piecewise functions, left- and right-hand limits are checked separately. Finally, Z3 can be used to produce a formal proof that the function is continuous at a given the established conditions.

When to Use It

• Determine if f is continuous at a given point a
• Verify continuity for piecewise-defined functions at boundary points
• Validate limits with symbolic computation using SymPy
• Obtain a formal proof of continuity with Z3 when needed
• Analyze potential discontinuities by inspecting left and right limits

Quick Start

1. Step 1: Identify the point a and confirm f(a) is defined
2. Step 2: Compute lim x->a f(x) with SymPy (and left-hand limit for piecewise if needed) using the command uv run python -m runtime.harness scripts/sympy_compute.py limit f(x) --var x --at a --dir left as needed
3. Step 3: If lim exists and equals f(a), optionally prove with Z3: uv run python -m runtime.harness scripts/z3_solve.py prove continuous_at_a

Best Practices

• Start by ensuring f is defined at a
• Compute lim x->a f(x) and compare with f(a)
• Use SymPy to automate limit calculations and algebraic checks
• For piecewise functions, evaluate left- and right-hand limits at the boundary
• When needed, formalize the result with a Z3 proof of continuous_at_a

Example Use Cases

• f(x)=sin(x)/x for x not equal 0 and f(0)=1
• f(x)=|x| is continuous for all real x
• f is x^2 for x<0 and x for x>=0 with f(0)=0
• f(x)=1/x for x not 0, f(0)=0 is not continuous at 0
• f(x)=floor(x) is not continuous at integers

Frequently Asked Questions