  To any compact $K\subset\hat{\mathbb{C}}$ we associate a map $\lambda_K:
\hat{\mathbb{C}}\rightarrow\mathbb{N}\cup\{\infty\}$ -- the lambda function of
$K$ -- such that a planar continuum $K$ is locally connected if and only if
$\Lambda_K(x)\equiv0$. We establish basic methods of determining the lambda
function $\lambda_K$ for specific compacta $K\subset\hat{\mathbb{C}}$,
including a gluing lemma for lambda functions and some inequalities. One of
these inequalities comes from an interplay between the topological difficulty
of a planar compactum $K$ and that of a sub-compactum $L\subset K$, lying on
the boundary of a component of $\hat{\mathbb{C}}\setminus K$. It generalizes
and quantifies the result of Torhorst Theorem, a fundamental result from plane
topology. We also find three conditions under which this inequality is actually
an equality. Under one of these conditions, this equality provides a
quantitative version for Whyburn's Theorem, which is a partial converse to
Torhorst Theorem.
