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We prove that $C_{\infty}(X)$ is an ideal in $C(X)$ if and only if every open locally compact subset of $X$ is bounded. In particular, if $X$ is a locally compact Hausdorff space, $C_{\infty} (X)$ is an ideal of $C(X)$ if and only if $X$ is a pseudocompact space. It is shown that the existence of some special functions in $C_{\infty} (X)$ causes $C_{\infty}(X)$ not to be an ideal of $C(X)$. Finally we will characterize the spaces $X$ for which $C_{\infty}(X)$ and $C_K(X)$, or $C_{\psi} (X)$, coincide.
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