WordPress at a glance
function is not described

wp_customize_support_script() WP 3.4.0

Prints a script to check whether or not the Customizer is supported, and apply either the no-customize-support or customize-support class to the body.

This function MUST be called inside the body tag.

Ideally, call this function immediately after the body tag is opened. This prevents a flash of unstyled content.

It is also recommended that you add the "no-customize-support" class to the body tag by default.

No Hooks.

Return

Null. Nothing.

Usage

wp_customize_support_script();

Changelog

Since 3.4.0 Introduced.
Since 4.7.0 Support for IE8 and below is explicitly removed via conditional comments.

Code of wp customize support script: wp-includes/theme.php WP 5.2.2

<?php
function wp_customize_support_script() {
	$admin_origin = parse_url( admin_url() );
	$home_origin  = parse_url( home_url() );
	$cross_domain = ( strtolower( $admin_origin['host'] ) != strtolower( $home_origin['host'] ) );

	?>
	<!--[if lte IE 8]>
		<script type="text/javascript">
			document.body.className = document.body.className.replace( /(^|\s)(no-)?customize-support(?=\s|$)/, '' ) + ' no-customize-support';
		</script>
	<![endif]-->
	<!--[if gte IE 9]><!-->
		<script type="text/javascript">
			(function() {
				var request, b = document.body, c = 'className', cs = 'customize-support', rcs = new RegExp('(^|\\s+)(no-)?'+cs+'(\\s+|$)');

		<?php	if ( $cross_domain ) : ?>
				request = (function(){ var xhr = new XMLHttpRequest(); return ('withCredentials' in xhr); })();
		<?php	else : ?>
				request = true;
		<?php	endif; ?>

				b[c] = b[c].replace( rcs, ' ' );
				// The customizer requires postMessage and CORS (if the site is cross domain)
				b[c] += ( window.postMessage && request ? ' ' : ' no-' ) + cs;
			}());
		</script>
	<!--<![endif]-->
	<?php
}