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.