Filters 'excerpt_more' in WordPress to provide a 'Continue reading' link. Add to your theme's functions.php file.

custom_excerpt_more.php

<?php

/**
 * Filters 'excerpt_more' to provide a 'Continue reading' link.
 */
function custom_excerpt_more($more) {
    global $post;
    $more = '&hellip;. <a href="'. get_permalink($post->ID) . '">Continue reading.</a>';
    return $more;
}

add_filter('excerpt_more', 'custom_excerpt_more');