/continuity

name: continuity
description: "Problem-solving strategies for continuity in real analysis"
allowed-tools: [Bash, Read]

Continuity

When to Use

Use this skill when working on continuity problems in real analysis.

Decision Tree

1. **Check Definition**

• f(a) exists (function defined at point)
• lim_{x->a} f(x) exists
• lim_{x->a} f(x) = f(a)

2. **Use SymPy for Limit Check**

• `sympy_compute.py limit "f(x)" --var x --at a`
• Compare with f(a)

3. **Piecewise Functions**

• Check left and right limits separately
• `sympy_compute.py limit "f(x)" --var x --at a --dir left`

4. **Verify with Z3**

• `z3_solve.py prove "limit_exists implies continuous"`

Tool Commands

Sympy_Limit

uv run python -m runtime.harness scripts/sympy_compute.py limit "f(x)" --var x --at a

Sympy_Limit_Left

uv run python -m runtime.harness scripts/sympy_compute.py limit "f(x)" --var x --at a --dir left

Z3_Prove

uv run python -m runtime.harness scripts/z3_solve.py prove "continuous_at_a"

Cognitive Tools Reference

See `.claude/skills/math-mode/SKILL.md` for full tool documentation.