Why do most function images take the form of a curve? I always see curves (such as in dark blue below) and not any other type of (function) image. May I get a reason for why it is always like that?

This figure is only for representation.
Your final line

This figure is only for representation.

pretty much answers your question.
When explaining these concepts you want to draw something that is easy to understand, i.e. does not have too many irregularities but is also not too regular.
Linearity is a special attribute of functions. General curves show that there is usually no need for this assumption.
Continuity is a frequent though not omnipresent assumption in economics.
Rather than throw in a few points of discontinuity and provide reasons for them, it is easier to explain general concepts with continuous curves. This does not mean that most functional relationships are indeed continuous; there are plenty of counterexamples, but where this is important, there are usually discontinuous graphs and explanations thereof.