Only show theme option if it option exists

If you’re wanting to output based on the option being defined, then use isset(). e.g. this:

<?php echo $up_options->slideshow6; ?>

…becomes this:

<?php 
if ( isset( $up_options->slideshow6 ) ) {
    echo $up_options->slideshow6;
}
?>

EDIT

How would I get the following to appear only if “slideshow6” was defined <a href="#"><img src="<?php echo $up_options->slideshow1; ?>" alt="#"/></a>

I would go with:

<?php 
if ( isset( $up_options->slideshow6 ) ) {
    echo '<a href="#"><img src="' . $up_options->slideshow1 . '" alt="#"/></a>';
}
?>