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The way a human brain reflects reality and the way the universe actually is are two different things. Through our history, we became good at detecting patterns. We learned that some visual images mean that we are moving or that some objects are farther than others. These conclusions are usually true, but not always: this is how visual illusions work. One of my favorite is this.
Similarly, in math, we often use intuition. We know through experience that some mathematical objects have certain properties, and we jump to conclusions without showing every step rigorously (at least, at first). However, the reality of mathematical objects is different. Sometimes we even have "mental illusions" where we expect that some object would behave differently from how it actually does.
To clear potential confusion, we need to try to suspend our intuition and work with precise definitions. Intuition is great for directing us when we search for a proof because typically there are too many possible paths and one can't examine them all. However, when we are talking about the most basic definitions, such as that of a continuous function, we have make sure that our intuition is accurate and to start with a clean slate, if necessary.
To begin with, a function is not a curve on a plane but a set of pairs of real numbers. Second, there is a precise definition of when a function is continuous. It has nothing to do with points being connected by straight lines. Third, Weierstrass function may contradict your idea about "the components of a continuous function", but it is not a mathematical contradiction. To know that something is indeed wrong, you need to come up with a much more concrete contradiction. For example, if you can rigorously show that the existence of the Weierstrass function implies that 0 = 1, that would indeed mean that such function cannot exist.
So, to understand why the Weierstrass function is continuous, you have to start with precise definitions and work through the proof.