TITLE: How do I tell if I am able to go to graduate school in math?
QUESTION [24 upvotes]: This is my first question on this site, and this question may sound disturbing. My apologies, but I truly need some advice on this.
I am a sophomore math major at a fairly good math department (top 20 in the U.S.), and after taking some upper-level math courses (second courses in abstract algebra and real analysis, differential geometry, etc), I can say that I genuinely like math, and if I have A BIT chance to succeed, I will go to graduate school and choose math research as my career.
However, this is exactly the thing that I am afraid of. My grades on the courses are mediocre (my GPA for math courses is around 3.7), and for the courses I got A's, I had to work very hard, much harder than others to get the same result, and I often get confused in many of the classes, while the others understand the material quickly and could answer professor's questions, and at the same time I didn't even understand what the professor was really asking. I really wonder, if I have to work hard even on undergraduate courses, does that mean I am not naturally smart enough for more advanced math, especially compared to everyone else in my class? Can I even survive graduate level math if I even sometimes struggle with undergraduate courses? I always believe that adequate mathematicians could do well in their undergraduate courses easily. In my case, even if I work very hard, I forget definitions/theorems easily and then of course forget how to use them to solve problems.
Is it still worth to try if I am significantly behind the regular level and have to work hard even for undergraduate courses, providing that there are a lot of smart people who can understand them instantly. This feeling hurts me a lot, especially when I am struggling with something in math, I always feel I am a useless trash and ask myself why I am so stupid?
I thought about talking to my professors about this issue, but I find this too embarrassing to start. I am really afraid that if I ask them this question, they may tell me the truth in person that "you are really not smart enough to go to graduate school".
So how can I tell if it is still worth for me to think about this path, or I should realize that I have no chance to succeed and give up now? I appreciate encouraging comments, but please, please be honest on this case because it is really important for my future plan. Thanks again for your advice, and I am really grateful.

REPLY [2 votes]: The answers here are rather idealistic. They seem to be based more on trite cliches rather than concrete reasoning or evidence. 
The fact is, academia is competitive and jobs are scarce. Your grades matter.  Your performance relative to your peers matters. Being passionate at math or being interested in the subject is a necessary but insufficient condition to succeed in graduate school. 
I'm going to be brutally honest with you: do yourself a favour and don't pursue graduate school. I'm sorry if that's your dream, but we must have a sense of realism. It's nothing more than several years of daunting work and, in return, you get to call yourself a "mathematician". Job prospects? Salary? Unexceptional, if you're lucky. 
Learn to program, and get a job in that field; people who are strong at math are almost invariably good at programming. If you're struggling and genuinely think you're incapable, I'm not going to lie to you and say that passion and perseverance is necessarily going to fix everything; it might, of course, but that's a conclusion you must reach, perhaps with the help of your professors. 
I'm not saying any of this because I think you lack the talent; I'm saying it because it's the advice I'd give to any friend, unless he or she is genuinely a genius. Indeed, I'm making this same decision myself. 
This answer may well be down-voted, and that's fine; it's simply a consequence of this site's demographics. Responses to this sort of question will invariably be opinionated, but, nevertheless, it is an important one.