Let

$X$,

$Y$ be realcompact spaces,

$E$ and

$F$ normed spaces,

$C(X,E)$ the set of all continuous

$E$-valued functions on

$X$ and

${C}_{b}(X,E)$ the space of all bounded continuous functions from

$X$ into

$E$. A linear map

$T:C(X,E)\to C(Y,F)$ is called separating if

$\parallel f\left(x\right)\parallel \phantom{\rule{4pt}{0ex}}\parallel g\left(x\right)\parallel =0$ for every

$x\in E$ implies that

$\parallel \left(Tf\right)\left(y\right)\parallel \phantom{\rule{4pt}{0ex}}\parallel \left(Tg\right)\left(y\right)\parallel =0$ for every

$y\in Y$, and biseparating if

${T}^{-1}$ exists and is separating as well. One result of the paper is that every linear biseparating map

$T:{C}_{b}(X,E)\to {C}_{b}(Y,F)$ is continuous provided that

$Y$ has no isolated points. Another result tells us that if additionally

$E$ and

$F$ are infinite-dimensional, then a linear biseparating map

$T:C(X,E)\to C(Y,F)$ is continuous if the interior of the set of

$P$-points of

$Y$ is empty. This is the best possible result.