[syntax] Can typescript export a function?

It's hard to tell what you're going for in that example. exports = is about exporting from external modules, but the code sample you linked is an internal module.

Rule of thumb: If you write module foo { ... }, you're writing an internal module; if you write export something something at top-level in a file, you're writing an external module. It's somewhat rare that you'd actually write export module foo at top-level (since then you'd be double-nesting the name), and it's even rarer that you'd write module foo in a file that had a top-level export (since foo would not be externally visible).

The following things make sense (each scenario delineated by a horizontal rule):

// An internal module named SayHi with an exported function 'foo'
module SayHi {
    export function foo() {
       console.log("Hi");
    }

    export class bar { }
}

// N.B. this line could be in another file that has a
// <reference> tag to the file that has 'module SayHi' in it
SayHi.foo();
var b = new SayHi.bar();

file1.ts

// This *file* is an external module because it has a top-level 'export'
export function foo() {
    console.log('hi');
}

export class bar { }

file2.ts

// This file is also an external module because it has an 'import' declaration
import f1 = module('file1');
f1.foo();
var b = new f1.bar();

file1.ts

// This will only work in 0.9.0+. This file is an external
// module because it has a top-level 'export'
function f() { }
function g() { }
export = { alpha: f, beta: g };

file2.ts

// This file is also an external module because it has an 'import' declaration
import f1 = require('file1');
f1.alpha(); // invokes f
f1.beta(); // invokes g