How to call action function in javascript remoting

Here is an exaple 

<apex:page controller="DF12RemotingController" docType="html-5.0" showHeader="false" standardStylesheets="false">
<apex:styleSheet value="{!$Resource.AdvVFWebinarCSS}"/>
<apex:composition template="DF12Template">
    <apex:define name="pageTitle">DF '12: JavaScript Remoting Example</apex:define>
    <apex:define name="header">
        <h1 class="pageTitle">Find Customer</h1>
        <h4>JavaScript Remoting Example</h4>
    </apex:define>
    <apex:define name="body">
        <input id="searchField" type="text" placeholder="Enter Last Name"/>
        <button type="button" onclick="console.log('Step 1: Search button clicked.');DF12Examples.handleButtonClick();">Search</button>
        <table cellpadding="0" cellspacing="0">
            <thead></thead>
            <tfoot></tfoot>
            <tbody id="results"></tbody>
        </table>
        <div class="cheatLinkSeparator">
            <a class="cheatLink" href="{!$Page.DF12Action}" title="Open apex:actionFunction Example">apex:actionFunction Example</a>
        </div>
    </apex:define>
    <apex:define name="pageScripts">
        <script>
            // startTime and endTime are only being created for logging purposes, but
            // note that I am initializing their values to Date objects. this is so
            // JavaScript doesn't have to guess what kind of variables they are
            // (and therefore, doesn't have to change their datatype mid-execution).
            // this is a best practice for all JS variable declarations to help with
            // performance.
            DF12Examples.startTime = new Date();
            DF12Examples.endTime = new Date();
            // gets called when the 'Search' button is clicked
            DF12Examples.handleButtonClick = function() {
                console.log('Step 2: JS function handleButtonClick() called.');
                DF12Examples.startTime = Date.now();
                var searchTerm = document.getElementById("searchField").value;
                // call the apex controller method tagged w/ the @RemoteAction annotation.
                // the syntax is ControllerName.methodName(methodArguments, jsCallbackFunctionName, options).
                // the 'options' argument is optional.
                console.log('Step 3: JS function that maps to Apex controller method doSearch(String customerLastName) - via @RemoteAction annotation - called.');
                DF12RemotingController.doSearch(searchTerm, DF12Examples.renderResults);
            };
            // this is the js callback function that will execute when the @RemoteAction method
            // in the apex controller is done executing. it will always receive two arguments
            // passed from the controller method: 1) the JSON-encoded results, and an 'event'
            // object, which tells you if everything went ok on the server side.
            DF12Examples.renderResults = function (results, event) {
                console.log('Step 4: Results recieved. JavaScript function renderResults() called.');
                // if the apex method completed execution, event.status is true.
                if (event.status) {
                    // if there was an exception, you should handle it here.
                    if (event.type === "exception") {
                        // do stuff
                    } else {
                        // this is the empty <tbody> tag on the page
                        var container = document.getElementById("results"),
                                 html = [];
                        for (var i=0, j=results.length; i<j; i++) {
                            html.push("<tr><td>");
                            html.push(results[i].LastName + ", " + results[i].FirstName);
                            html.push("</td></tr>");
                        }
                        // this won't work in IE, because IE doesn't handle setting
                        // innerHTML for <tr> or <tbody> tags. *sigh*. To get this
                        // working in IE, you'd need to use document.createElement()
                        // for every DOM node you want to create, then use appendChild()
                        // to attach them to the correct parent.
                        container.innerHTML = html.join("");
                    }
                }
                DF12Examples.endTime = Date.now();
                if (console) {
                    console.log("elapsed time: " + (DF12Examples.endTime - DF12Examples.startTime) + "ms");
                }
            };
        </script>
    </apex:define>
</apex:composition>
</apex:page>

Class 

public with sharing class DF12RemotingController {
    @RemoteAction
    public static List<Contact> doSearch(String customerLastName) {
        // be careful when doing LIKE queries with '%' wildcards on both
        // sides of the query term, especially when using non-indexed
        // fields, as performance can suffer
        customerLastName = '%' + customerLastName + '%';
        return [
            SELECT Id, FirstName, LastName 
            FROM Contact 
            WHERE LastName LIKE :customerLastName 
            ORDER BY LastName, FirstName 
            LIMIT 200
        ];
    }
    @isTest
    private static void testDoSearch() {
        List<Contact> control = new List<Contact>();
        control.add(new Contact(LastName = 'Smith', FirstName = 'Joe1', Email = 'test@test.test.test.test'));
        control.add(new Contact(LastName = 'Smith', FirstName = 'Joe2', Email = 'test@test.test.test.test'));
        insert control;
        List<Contact> actual = DF12RemotingController.doSearch('Smith');
        System.assert(actual.size() == 2, 'There should be two contacts named Smith here.');
    }
}