I think every time I teach about anatman I get some variation on this question. Most of the questions arise from a misunderstanding of the term and meaning of ‘anatman’. The Buddha lived in a time and place where the soul - termed atman in Sanskrit - was considered to be an eternal, unchanging expression of an eternal, unchanging God (Brahma). The goal of “enlightenement” in the Hinduism of the time was to reunite this personal atman with Brahma.
In his time under the Bodhi Tree, as the Buddha became the Buddha, he realized not only the causes and reliefs of dissatisfaction but also the impermanence of the self. This was not because the self does not exist - although there are schools of Buddhism that teach that - but because it is neither eternal nor unchanging. We - the collection of selfs reading this - are an interconnected, interdependent part of the Universe and all things compounded in the Universe are subject to change and passing away.
I’ve discussed elsewhere the concept of Dependent Origination - that all things arise from causes. This is true of the self as well. Some of those causes are external - our parents, friends, education, genetics and epigenetics, etc. - and some are internal - our own decisions and actions as we walk through life. Whatever the causes are they aren’t once and done. They continue throughout our lives. This alone shows that our self is not unchanged. The self of yesterday - and the influences on it - gives rise to the self today who will in turn give rise to the self of tomorrow; but none of these selves are exactly the same.
The impermanence of the self is more difficult to show since our perceptions are confined to this reality here and now. We have no evidence of the flickering out of the self; but we also have to evidence of the eternity of the self. We have to draw inferences from the things that we can observe, and what we observe is that there is no compounded thing that we can observe in the Universe that is eternal. As a part of the Universe we must accept that the same rules apply to us as well.
Buddhism, however, gives precedence to science. Even the Dalai Lama has said that if science proves a teaching to be untrue we must accept it and change our teaching. If science were to prove the eternity of the self - or even its survival of death - we would change this teaching. For the moment though the important question to ask is, “Is it more useful to see the self as eternal or to see it as not?”
The Buddha’s teaching is about the cessation of suffering. In the Theravadan tradition this is thought to be carried out over many rebirths, but Mahayana brought in the thought that we need to focus on this life and do what we can to alleviate dissatisfaction for all being in this life. This is the lineage of my ordination and thought. By focusing on the self in this life and this life alone we are giving the motivation to do the best you can in this life - both for yourself and others. In the Kalama Sutra the Buddha explains the reasons for acting with good will, compassion, equanimity and appreciation in this way:
"'If there is a world after death, if there is the fruit of actions rightly & wrongly done, then this is the basis by which, with the break-up of the body, after death, I will reappear in a good destination, the heavenly world.' This is the first assurance he acquires.
"'But if there is no world after death, if there is no fruit of actions rightly & wrongly done, then here in the present life I look after myself with ease — free from hostility, free from ill will, free from trouble.' This is the second assurance he acquires.’”
Since any heavenly reward cannot be shown to be real by our observations in this life the most skillful way to act is in terms of the second assurance - that we might be “free from hostility, free from ill will, free from trouble”. This is the way of anatman. This is the way of the Pragmatic Buddhist.