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.