So, I am taking introduction to real analysis next semester, and I heard that it may be a challenging course. But what I want to know is the applications of real analysis, which I imagine there are some of, considering how the phrase "real analysis" seem to be thrown around quite a bit. Thanks.
You really should be clearer about what you mean by "applications". Certainly one "applies" theorems from functional analysis to show that certain differential equations have solutions with certain properties. That's an application to mathematics rather than physics but it is still an application. Since mathwonk mentioned existence theorems for differential equations, here is a clever "physics" application of the "uniqueness theorem" for differential equations. Suppose you have a taut string, attached to a point on a wall at one end, the other end in your hand. You flip an upward "hump" in the string that moves down the string to the wall. Obviously when it hits the wall, it will "reflect" and come back toward you, but will it come back above or below the string? To answer that, imagine the string extending an equal distance on the other side of the wall with another "hump" moving toward the wall. That situation, with the two humps moving toward each other is a solution to the "wave equation". But in order that the wire not move (its fastened to the wall remember) when the two humps meet at the wall, they must cancel, not add. That means the "other hump" must be below the imaginary part of the string. But that "two hump" solution to the wave equation satisfies exactly the same conditions as our "one hump" solution on our side of the wall. Since such a solution is unique, the hump must come bake beneath the string.