I'll be the devils advocate. Why? I can understand calculus perfectly fine without understanding how people got there. In fact, I'd argue much of what you learn before call--in the order of discovery--actively hinders understanding.
However if all you care is to use what has been discovered, which is becoming less and less valuable, then you don't need to learn history of mathematics and science.
Although this is not good evidence but rather an anecdote, I cannot remember any significant person who has made fundamental contributions to mathematics or science that was completely ignorant of the history of the field.
For Euler, continuous function was pretty much intuitive notion. He only composed functions with only occasional point discontinuities, so it wasn't a big deal for him to even not have a proper definition.
Then people like Bolzano and Dirichlet came along and realized they need a better definition, because there can be some really weird cases. So they formalized the continuity with limits (which is typical way how to define it in basic calculus).
Later yet, people understood better what it means to be a real number by looking at notions such as countability and measurability. While this doesn't affect continuity itself, it does affect understanding of what is a real fuction.
Then came more abstraction, to metric spaces and eventually topological spaces, which redefined "continuous function" yet again as a morphism between topological spaces.
Another shift in thinking about continuity happened when theory of distributions was invented. This actually completely reverses the intuition - instead of properly definining reals and then on top of real function define what it means to be continuous, you define the "function" itself in an entirely different way, in which the continuity becomes somewhat irrelevant.
Finally, modern mathematics is quite obsessed with category theory and various ways to make everything into some algebra. In a way, we care less what reals really are, only what we can do with them (or their sets).
So I think to understand the intuitive relation of all these different definitions, you need to understand a little bit of history.
Also, mathematical concepts don't come with natural names attached. But we need consistent labels for successful communication. It's much easier to not confuse those labels if you know a bit about the history that led to the naming.