JSFiddle - React, Tailwind, and code Playground

JavaScript

function doSomething(timer) {
  timer = timer || 1000;
  var alreadyFired;
  var timerID = null;
  var interruptCounter = -100;
  var fireLimit = 5;
  var fired = 0;

  function Debounce() { //Old way, how I understood you inititally
    if (alreadyFired != true) {
      alreadyFired = true;
      //Do stuff
      console.log("fired");
      setTimeout(function() {
        alreadyFired = false;
      }, 1000); // Debouncing on 1 second
    }
  }

  function ConstantFiring(firstIndicator) { // New way, as per comments.
    //Do stuff
    if (firstIndicator === true) {
      console.log("fired for the 1st time");
    } else {
      console.log("Firing constantly every " + timer / 1000 + " seconds.");
    }
  }

  function Throttle() {
    if (fired < fireLimit) {
      fired++;
      //Do Stuff
      console.log(fired);
      setTimeout(function() {
        fired--;
      }, 3000); //Throttling on 5 fires per 3 seconds
    }
  }

  function InterruptTimer() {
    if (interruptCounter === -100) {
      //This runs only the 1st time window is resized, if you leave this part out, the function will only fire 3 seconds AFTER the first event fired and then consistantly unless the event occured in last 3 seconds.
      interruptCounter = 0;
      ConstantFiring(true);
    }
    if (interruptCounter <= 0) {
      interruptCounter++;
      clearInterval(timerID);
      timerID = null;
      setTimeout(function() {
        interruptCounter--;
        if (interruptCounter <= 0 && timerID == null) {
          timerID = setInterval(ConstantFiring, timer);
        }
      }, 3000); // Interrupt timer
    }
  }

  //Change the following line to use a different function to test.
  var realFunction = InterruptTimer;
  return realFunction;
}

window.onresize = doSomething(768); //Add a parameter representing a period for the function timer (in milisec) This is very optional, you can also hardcode it into the doSomething fucntion in case you want to add some other event.[parameter]...