Characterizations of the Bouligand contingent cone to the epigraph and graph of a locally Lipschitz function are given. It is proved that, if f:H\(\mapsto {\mathbb{R}}\) is a locally Lipschitz function on a Hilbert space H, then \(T_{Epi(f)}(x,f(x))=\{(x',y')| \quad \exists \{a_ n\}^{\infty}_{n=1},\quad \{y_ n\}^{\infty}_{n=1}\subset {\mathbb{R}},\quad \{x_ n\}^{\infty}_{n=1}\subset L(x,x'),\quad x_ n\to x,\quad \lim_{n\to \infty}a_ n((x_ n,y_ n)- (x,f(x)))=(x',y')\}.\) If \((x',y')\in T_{Graph(f)}(x,y)\) then \((x_ n,y_ n)\in Graph(f)\). (Here \(L(x,x'):=\{y\in H| y=x+t(x'-x),\quad t\in {\mathbb{R}}\}).\) Using this characterization, relations between the Bouligand contingent cone to the graph of a Lipschitz function and Dini derivatives are studied.