  For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of
real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open
topology and the pointwise topology, respectively. If $X$ is compact, the
classic result of A.~Grothendieck states that $C_k(X)$ has the Dunford-Pettis
property and the sequential Dunford--Pettis property. We extend Grothendieck's
result by showing that $C_k(X)$ has both the Dunford-Pettis property and the
sequential Dunford-Pettis property if $X$ satisfies one of the following
conditions: (i) $X$ is a hemicompact space, (ii) $X$ is a cosmic space (=a
continuous image of a separable metrizable space), (iii) $X$ is the ordinal
space $[0,\kappa)$ for some ordinal $\kappa$, or (vi) $X$ is a locally compact
paracompact space. We show that if $X$ is a cosmic space, then $C_k(X)$ has the
Grothendieck property if and only if every functionally bounded subset of $X$
is finite. We prove that $C_p(X)$ has the Dunford--Pettis property and the
sequential Dunford-Pettis property for every Tychonoff space $X$, and $C_p(X) $
has the Grothendieck property if and only if every functionally bounded subset
of $X$ is finite.
