Can extra virgin olive oil be used for stir frying, roasting, grilling?

That oils' smoke points can be generically classified solely according to their type is a myth.

Robert Wolke, a professor emeritus of chemistry at the University of Pittsburgh and food columnist for the Washington Post, claims that the smoke point for an oil varies widely depending on origin and refinement. While the smoke point does generally increase with respect to the degree of refinement of the oil, Wolke notes that generation of the nasty tasting free fatty acids is a function of time, i.e., the longer you cook the oil the worse it tastes. Therefore, if you are going to be using the oil for a quick fry/sauté, it doesn't really matter what type of oil it is. (I'm not sure if I'd use an unrefined oil for a long roast, though.)

Alton Brown, a US cooking celebrity who specializes in the science behind cookery, agrees:

Now many charts and tables attempt to quantify smoke points, and I'm here to tell you they're all complete hooey. The truth is, there are just too many factors going into a smoke point to make such concrete claims. I will tell you this. High heat will destroy the fruity goodness of an extra virgin olive oil or the nutty goodness of a walnut oil. But you can sauté with just about any oil, as long as you work fast.
_{[Emphasis is mine.]}

I wouldn't recommend using an Extra Virgin Olive Oil for high heat applications, not because you're likely to get off flavors (if you work fast), but because it's a waste of money: You'll be losing the fruity notes for which you're paying a premium. Throughout Southern Italy and Spain, though, it is very common to use olive oil for frying (even Virgin olive oil). For example, Mario Batali (another US celebrity chef who is a widely acclaimed Italian chef) did it all the time in his old TV show.

I don't want to disappoint you, but the sad truth is that extra virgin olive oil is unsuitable for all the cooking methods you mention. When you heat any oil past its smoking point it starts to deteriorate and can even become dangerous. Olive oil, extra virgin in particular, has a lower smoking point than most other oils. In fact, you will be better off with an olive oil of lesser quality. Such an oil will have been processed and thus purified, increasing its smoking point.

You may think that butter is unhealthy, but at high temperatures it will actually withstand the heat better than extra virgin olive oil.

Good oils for the purposes you mention are canola oil, sunflower oil, peanut oil, and grapeseed oil.

Aside from agreeing with ESultanik, I wanted to add this additional answer:

Virgin olive oil is not appropriate for stir-frying because it would taste weird.

Unless you're doing a recipe which is specifically an Italian/Chinese fusion dish, or something which expects olive oil, you want to use a mostly flavorless vegetable oil for stir-frying. I prefer peanut oil, but canola, safflower, and sunflower also work quite well.

I have used olive oil for all of those tasks when I didn't have anything else around. It's also my preferred oil for oven-roasting vegetables. As Henrik says, though, it has a low smoke point, so it doesn't work as well as some others for cooking in a wok, or other high temperature cooking.

Mario Batali does it all the time, so it's safe to say you can use it for that. Should you? Probably not, but if the $$$ doesn't bother you...

Referring to a series of articles written on cooking at high heat with olive oil: The answer is yes, it's perfectly fine to use olive oil to cook at high heat.

Reasons: 1. "Olive Oil is High in Monounsaturated Fats, Which Are Stable When Heated"

2. "Extra Virgin Olive Oil is High in Antioxidants and Vitamin E, Which Help Fight Oxidation (Olive oil contains Vitamin E and many powerful antioxidants. These substances protect the oil from damage during high heat cooking.)"

The reasons normally stated that advised against it is based on the fact of nutrition loss --> Again, nutrition is lost when you cook vegetables as well (Doesn't stop me from cooking them)

References: https://authoritynutrition.com/is-olive-oil-good-for-cooking/

https://healthimpactnews.com/2014/myth-buster-olive-oil-is-one-of-the-safest-oils-for-frying-and-cooking/

www.seriouseats.com/2015/03/cooking-with-olive-oil-faq-safety-flavor.html

www.oliveoiltimes.com/whats-cooking-with-olive-oil/frying-with-olive-oil/30470

You have to consider that extra virgin olive oil usually has a smoking point around 190 degrees celsius. For stir frying and getting the natural sugar of e.g. the vegatables to to be extracted and caramelized you need around 150 degrees celsius. I use extra virgin olive oil for fryin vegetables and it maintains it's fruity flavor plus all of it's health benefits. Make a test on your stove to figure out your extra virgin olive oil's smoking point, and make sure to keep the temperature below the oil's smoking point. I use gas so it's very consistent, and I've made a mark on the regulator, so I don't have to worry about the oil overheating every time I fry something. Once you get used to frying you immediately smell if the oil is smoking.