continuity

# 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 ```bash uv run python -m runtime.harness scripts/sympy_compute.py limit "f(x)" --var x --at a ``` ### Sympy_Limit_Left ```bash uv run python -m runtime.harness scripts/sympy_compute.py limit "f(x)" --var x --at a --dir left ``` ### Z3_Prove ```bash 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.