Source

File: wp-admin/includes/class-wp-site-health-auto-updates.php

function test_all_files_writable() {
		global $wp_filesystem;

		require ABSPATH . WPINC . '/version.php'; // $wp_version; // x.y.z

		$skin    = new Automatic_Upgrader_Skin;
		$success = $skin->request_filesystem_credentials( false, ABSPATH );

		if ( ! $success ) {
			return false;
		}

		WP_Filesystem();

		if ( 'direct' !== $wp_filesystem->method ) {
			return false;
		}

		$checksums = get_core_checksums( $wp_version, 'en_US' );
		$dev       = ( false !== strpos( $wp_version, '-' ) );
		// Get the last stable version's files and test against that.
		if ( ! $checksums && $dev ) {
			$checksums = get_core_checksums( (float) $wp_version - 0.1, 'en_US' );
		}

		// There aren't always checksums for development releases, so just skip the test if we still can't find any.
		if ( ! $checksums && $dev ) {
			return false;
		}

		if ( ! $checksums ) {
			$description = sprintf(
				/* translators: %s: WordPress version. */
				__( "Couldn't retrieve a list of the checksums for WordPress %s." ),
				$wp_version
			);
			$description .= ' ' . __( 'This could mean that connections are failing to WordPress.org.' );
			return array(
				'description' => $description,
				'severity'    => 'warning',
			);
		}

		$unwritable_files = array();
		foreach ( array_keys( $checksums ) as $file ) {
			if ( 'wp-content' === substr( $file, 0, 10 ) ) {
				continue;
			}
			if ( ! file_exists( ABSPATH . $file ) ) {
				continue;
			}
			if ( ! is_writable( ABSPATH . $file ) ) {
				$unwritable_files[] = $file;
			}
		}

		if ( $unwritable_files ) {
			if ( count( $unwritable_files ) > 20 ) {
				$unwritable_files   = array_slice( $unwritable_files, 0, 20 );
				$unwritable_files[] = '...';
			}
			return array(
				'description' => __( 'Some files are not writable by WordPress:' ) . ' <ul><li>' . implode( '</li><li>', $unwritable_files ) . '</li></ul>',
				'severity'    => 'fail',
			);
		} else {
			return array(
				'description' => __( 'All of your WordPress files are writable.' ),
				'severity'    => 'pass',
			);
		}
	}