• You have to show all your work, including intermediate steps, and explain how you reached your conclusion.

A. Complements and Applications

1. Consider the function f(x) = {x x rational

0 x irrational

(a) Show that f is continuous at x = 0

(b) Show that f is discontinuous at any x=0 (Hint: any open interval (a, b) contains both rational and irrational numbers)

2. (Physics). Special relativity shows that an observer will observe that the length of an object depends on the relative velocity v that the object is moving with respect to the observer, according to the law (Lorentz contraction)

L(v)=L₀1c2v2

where L₀ is the length observed when the object is at rest relative to the observer, and c is the speed of light (approximately 3 10⁸m/s, the same irrespective of the relative motion of the observer and the light source). Find

lim _vcL(c)

Why did we restrict to the left limit?