Thread: How can Weierstrass functions by defined by their definition without contradiction?

I'm just wondering on how a Weierstrass function can be defined by
it's definition and constitute as a continuous function without contradicting the
components of a continuous function?

What I mean exactly is, how can a Weierstrass function be composed of
a series of points (like any regular function) but not have any
"smooth" lines connecting from one point to another; no matter how
closely you observe the points of the function?

I thought a continuous function generally is (graphically) defined by a series of points, in which each point is connected by a straight line from the preceding point.

An example of a Weierstrass function I recently saw is: The summation
from "n = 0" to "infinity" of: "(1/2^n)*(sin((2^n)*x))".

Please help me wrap my head around this.

Welcome to the forum.

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.

Well then let's just approach the part that is confusing me directly; and start defining truths / facts in regards to continuous functions.

1) Are all points of a continuous function connected by "something"?

1a] If so, is this "something" always a line?
1a-i] If this "something" is not always a line, what else can connect points in a continuous function?

1b] If not, can a continuous function be a set of points that are infinitely close to each other, but are not connected by anything?

Thanks,
-Jshine

1) Are all points of a continuous function connected by "something"?

The definition of a continuous function does not say anything about connecting points. In fact, it would be a bit strange if it did. Indeed, the connections themselves have to be parts of the function's graph. Then some points are distinguished and the function's values corresponding to them are presumably given explicitly, while the values corresponding to the intermediate points are obtained from the connections that join the main points. But where does this distinction between main and intermediate points come from? In practice, functions are often given this way, but this is not the only way to specify a function, let alone the definition of a function.

1b] If not, can a continuous function be a set of points that are infinitely close to each other, but are not connected by anything?

Yes, at least, informally speaking. Paraphrasing the definition, a function is continuous at if moving around changes only a little compared to .

Everything I thought I knew about calculus has been compromised.