Continuity

When to Use

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