Is it common for new SaaS products to be at a breakeven with just a dozen paying customers? Should I seek investment or be happy to be in the black?

Definitely DON'T go for funding at this stage. It would be just TOO COSTLY for you in terms of the stake you will have to give away for a limited amount.

Most likely at this stage, you don't even have a clear picture if you really need the investment. Have you mobilized all the cash you can on your own? All I mean absolutely all. You don't want to ask for money when you are saving some bucks just in case your idea doesn't work. Have you raised some from FFF (Family, Friends and Fools)?. Again, don't go to angels until you have covered your very basics with those.

All in all, most probably what is best for you is to grow this a bit more, get some proper biz metrics in place (churn, growth, revenue per user ...), experiment a bit with which marketing tools will work best for you and start a fire. When you have this fire started (i.e. you know, backed by data, what works and what doesn't), then YOU MAY buy some gasoline (i.e. ask for cash). Don't do it before.

Always keep this in mind that I read in some book:
"A FOOL AND HIS/HER EQUITY ARE SOON DEPARTED"

Pere

Congrats on finding a quick win. The decision to seek investment really depends on your performance to this point, scalability, market potential and what you expect to give up in terms of equity. If you can grow without outside investment, I'd recommend it. Think about cheap sources of labor - we use interns for dev work depending on their experience level and skills. PR/marketing can be done in other ways besides the cookie cutter corporate campaigns. Look to debt or other sources of financing before making a decision to give up equity.

I'd be happy to talk about your platform and possible options. Send me a message if you'd like to find a time to meet.

Hi! My name is Humberto, I'm consultant focused on Growth Strategy and Sales with an MBA and international clients.
You make it sound like is bad thing! Breaking even at this point is really good thing regardless of business, specially in a SaaS product. With that said, it sounds like you are doing sales cold calling yourself and negotiating and doing everything personally versus having an automated system.
What you would benefit from regardless of investment is having a plan on how you are going to get exposure, get leads, build a pipeline of potential interested clients, grow vertically and what approaches can be assumes for now to be sustainable. Once you have a set of objectives to follow you can begin crafting a strategy for marketing at different levels of expenditure. (which btw having this strategy plan can help you secure investment)

Look for someone who is willing to work on equity and minimal pay for their work on PR or to help you layout the fundamental strategies you need to fully understand your own business model (I know, you probably think you do already) but trust me your not even close and I say that with best intentions.

Answers to your questions are interrelated through your personal capabilities and personal growth objectives in the context of the demands of building an organization custom fitted to the opportunity's potential, hiring the right people to sit in the right seats and tolerance of prolonged uncertainty. Answer these and the rout to getting the right kind of money from the right kind of people is greatly clarified.
Will explain further if you wish.

Hi,

You are in a great position.

At the risk of sounding like a broken record, you really have a marketing problem. You have to position your company and pitch to reveal the hidden benefits of acquiring your products or even your company. It will not be easy, but it can be done. Then you have to show the potential clients that they are really just buying tons of money at a discount. For the marketing problem, follow my advice below or get someone who knows how.
So, here are several of my marketing solutions:
Bonus:
Here is $10,000 worth of information for free and in a nutshell. (each of these proven concepts could actually be worth millions of dollars in the right hands.
I like your idea. As a matter of fact, I will go so far as say, it is a great idea. So...don't give up. But don't keep beating your head against the wall either. You have to get over the wall.
You have to quit focusing on marketing and sales tactics and focus on a strategy. How do you to this? You have to come up with a really solid USP. Why would a customer choose you over others similar companies? Find the answer to this before you continue with your marketing strategy.
Secondly, you need an irresistible offer.
Thirdly, you need to give an unbelievable guarantee. By doing this and following the directions below on pre-launch and launch you should be a lot closer to getting real solid customers.
I didn't come close to inventing the pre-launch or the launch. Concert goers are very familiar with winning concert tickets by calling into radio stations or winning vip treatment, or back stage passes, lunch with a star, the list goes on.
Likewise, if you look at "professional wrestling," the whole fitting before the fight is just a pre-launch. PT Barnum was doing this for circus goers over 100 years earlier. And I can only imagine the pre-launch of the Romans for the Gladiator Fights.
In more recent history, every type of business from Retail Stores to Real Estate companies have used multiple pre-launch techniques. Believe me, tourists are bombarded with Condo deals when they visit Disney Land.
This is similar, but different from lead generation (I don't have the enough space to discuss this powerful technique here). But I use both of these techniques in my own businesses including offering a free information packed newsletter and encouraging my clients to move up my sales latter because it is best for them. Most do move up the sales latter as their ambition and drive increases. Some move all the way up from the very beginning. Both benefit from this, one just takes longer to receive the benefits. Others will never take a chance on becoming successful.
Okay, more to your pre-launch campaign for SaaS. Simply, come up with a taste of what you have, ask a serious question and answer it. At the end of the end of the first "answer and solution" set the potential client up with another problem that is very familiar to them. Tell them that you have the answer. Follow this technique several time. I believe most do this repetition 2 or 3 times, but a famous golfer has sent me literally dozens of how to videos in order for me to take the bait.
You might think that giving the answer to a solution makes your product less valuable and your opinion less valuable. If you think that, you would be wrong. Heck, look at what I have given out in this answer. My experience is if you give you will receive. That is if you know what to give, how much to give, and how to receive.
I am not trying to sell you on calling me. Most people frankly cannot afford it. Really, I am pretty busy with my businesses and consulting. However, I need more info before I could have a greater impact in helping you.
Most solutions involved this: Ask, Ask, Ask, then Ask again.
Concentrate on the 3 M's. There are actually 7, but 3 will do for now. These are Market, Message, and Media. They come in that order.
Who is your target market (customer, clients, buyers, users, etc.)?
Tailor your laser focused message for this target market.
What is the best media mix to get your message to that market?
Here's what you do...first, make it an offer that is so incredible that they cannot resist. Secondly, do all the work for them. Make it so easy to make the purchase now that they can do it virtually without effort. Thirdly, give them an incentive to act right now. Fourthly, offer an almost unbelievable guarantee. Fifth, offer a bonus for acting now. There are many other incredible steps, but these steps should help the novice to the professional sell anything.
Whether you are selling B2B or B2C, you have to focus on selling to only one person. You can actually sell to one person at a time while selling to millions at a time. They are one and the same. Don't get off track, what we call digital marketing selling is just selling in print. And that has not changed since Cluade Hopkins wrote "Scientific Advertising." Really long before he wrote the book.
The secret to success: I have had the pleasure of knowing and working with some of the biggest names in business, celebrities, actors, entrepreneurs, business people, and companies from startup to billion dollar operations. The number one reason for their success is doing what they know and love while doing it in new, creative, and innovative ways.
Ask, Ask, Ask. Have thick skin and learn from each "mistake." In a short while, the market will tell you what you need to do and who and what you need to ask. But get started now even if that just means asking a contact on LinkedIn.
While you are thinking, think big and think of something at least 1% better, newer, or different. And being cheaper is not a winning strategy.
Make decisions quickly and change decisions slowly..unless you are actually going off a cliff.
Remember these two 11 letter words...persistence and consistency. They are two of the most important tools ever invented.
Even better yet, remember my 411 Rule of Achievement – It consists of (4) eleven letter words for super achievement (also an 11 letter word). Here it is, my 411 Rule of Super Achievement:
Persistence and
Consistency can change even the smallest
Possibility into a big time
Probability
Persistence + Consistency =
Possibility  Probability
By the way, I get a lot of people asking me if I can take phone calls for free (a free sample). Sorry, I can’t. I respect Clarity.fm and what they are trying to accomplish.
Treat everybody you talk to and everybody you meet (including yourself) like each is your number one million dollar customer.
Remember this for most people who really want to achieve a dream:
First: Your dreams are important and those who don’t support and believe in your dreams either don’t understand your desire and ambition or they have some other reason (many times reasons they themselves don’t understand) for not wanting you to spend the time and effort necessary to achieve your dreams.
Secondly: If you haven’t achieved your dreams and goals so far, it is not your fault. I know that this goes against what you usually hear, but it is true. Stop blaming yourself. You have a whole world of obstacles that are truly the blame. You only need to figure out how to go over, go under, go through, go around, or go with these obstacles in the direction of your dreams.
Thirdly: Fear is normal, but don’t give into it. Use it to motivate you and guide you.
Fourth: You are right; there are probably some people who don’t want you to succeed.
Fifth: Keep this in mind, there are people competing to get their first, do more, have more, invent what you are considering inventing, or simply trying to win. Believing in yourself and what you are doing is part of a strong recipe for winning over your competition.
I also always suggest that everybody at every stage work with a coach, mentor, or consultant. Heck, it works for Tiger Woods, every team in the NFL, the NBA, and etc. We all need guidance and support.
Best of luck,
Take massive action and never give up.
Michael
Michael Irvin, MBA, RN
www.michaelvonirvin.com
PS – Many people have “Upvoted” my answers. Thanks to those who do this. I really appreciated.