Can a JavaScript object have a prototype chain, but also be a function?

function a () {
    return "foo";
}

a.b = function () {
    return "bar";
}

function c () { };
c.prototype = a;

var d = new c();
d.b(); // returns "bar"
d(); // throws exception, d is not a function

Is there some way for d to be a function, and yet still inherit properties from a?

Actually, it turns out that this is possible, albeit in a non-standard way.

Mozilla, Webkit, Blink/V8, Rhino and ActionScript provide a non-standard __proto__ property, which allow changing the prototype of an object after it has been created. On these platforms, the following code is possible:

function a () {
    return "foo";
}

a.b = function () {
    return "bar";
}

function c () {
    return "hatstand";
}
c.__proto__ = a;

c(); // returns "hatstand"
c.b(); // returns "bar"; inherited from a

This might be of use to anyone who doesn't need to worry about cross-platform compatibility.

However, note that only the properties of an object can be inherited. For example:

var d = {};
d.__proto__ = a;
d.b(); // returns "bar"
d(); // throws exception -- the fact that d is inheriting from a function
     // doesn't make d itself a function.

Short answer: not possible.

This line of your code:

var d = new c();

automatically assumes that d is an object. Unless c is a constructor of a builtin object, e.g., Function. But if c is already defined by the language, you cannot manipulate its prototype, and cannot "inherit" it from whatever you like. Well, in some interpreters you can, but you cannot do it safely across all interpreters — the standard says: "you doth not mess with standard objects or the interpreter will smite you!".

The builtin objects are "unique" and JavaScript does not provide ways to duplicate them. It is not possible to recreate String, Number, Function, and so on without resorting to incompatible trickery.

Based on a discussion on meta about a similar question I'm posting this answer here based on @alexander-mills original

This can now be done in a standards compliant way

First create an object which inherits Function

const obj = Object.create(Function.prototype);  // Ensures availability of call, apply ext

Then add you custom methods and properties to obj

Next declare the function

const f = function(){
    // Hello, World!
};

And set obj as the prototype of f

Object.setPrototypeOf(f,obj);

Demonstraction

const obj = Object.create(Function.prototype);

// Define an 'answer' method on 'obj'
obj.answer = function() {
  // Call this object
  this.call(); // Logs 'Hello, World'
  console.log('The ultimate answer is 42');
}

const f = function() {
  // Standard example
  console.log('Hello, World');
};

Object.setPrototypeOf(f, obj);

// 'f' is now an object with an 'answer' method
f.answer();
// But is still a callable function
f();

Yes, it is possible if you use the __proto__ property Daniel Cassidy mentioned. The trick is to have c actually return a function that has had a attached to its prototype chain.

function a () {
    return "foo";
}

a.b = function () {
    return "bar";
}

function c () {
    var func = function() {
        return "I am a function";
    };
    func.__proto__ = a;
    return func;
}
c.prototype = a;

var d = new c();
d.b(); // returns "bar"
d(); // returns "I am a function"

However, you'll need to do some more tweaking of the prototype chain if you want instanceof to return better results.

d instanceof c // true
d instanceof a // false
c instanceof a // false