Function Pathologies

Before blindly applying a root-finding algorithm to a problem, it is essential that the form of the equation in question be understood: graph it!
For smooth functions good algorithms will always converge, provided the initial guess is good enough.
Pathologies include discontinuities, singularities, multiple or very close roots, or no roots at all!

Before blindly applying a root-finding algorithm to a problem, it is essential that the form of the equation in question be understood: graph it!