JavaScript codepen

After playing around with the code a bit I identified two issues:

Firstly in this line:

var $input = $(".input");

You set $input to an element with a class called input but there is no such class. I assume you might have meant to set it to the element with the input tag instead, in that case you should not use . in the selector. Like this:

var $input = $("input");

Though that is only a stop-gap measure that works because there is only one input element on the webpage. If you want to add validation to other inputs in the future you have to actually add classes or id's to them and then use that to select them.

The second issues are these lines:

var $inputVal = $input.val();
var $falseInput = !$input.val();

Variables are not updated dynamically when the thing you set them to changes. They are assigned the value the function returns and then they just keep that value. Since you assign these variables at the top of your code they are assigned a value once the page loads and then just keep that value. Meaning that they are not updated when something is written into the input. These two lines should be placed within the click listener of the submit button, that way their value will always reflect the current state of the $input variable.

Here is the revised code I ended up writing:

var $errorMssg = $("#error-mssg");
var $input = $("input"); // Removed the . since "input" is not a class

$(".submit").click(function errorDisplay(e) { // Added event parameter since I don't want the page to refresh
  var $inputVal = $input.val(); // Moved these two lines down here
  var $falseInput = !$input.val(); // So they are run each time they are referenced

  if ($falseInput) {
    $errorMssg.show();
    e.preventDefault(); // I added this to prevent the page from reloading when the submit event was sent
  } else if (!$falseInput) {
    $errorMssg.hide();
    e.preventDefault();
  } else{
    return true;
  }
});

I added the e.preventDefault(); mostly to make debugging easier on myself, but It's probably a good idea to keep in your code while you are experimenting around with the submit event handler.

If you have any additional questions, or found part of my explanation unclear then feel free to ask followup questions, I'll be happy to answer any question I can.