Consider the function defined by f(x)=|x|

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

Consider the function

f:\mathbb R\to\mathbb R

defined by

f(x)=|x|\,\lfloor x \rfloor.

Answer the following questions. (i) State the formal definition of continuity of a real-valued function at a point

x=c

. (ii) Using the definition, examine the continuity of

f

x=0

. Give full justification for your answer. (iii) Examine the continuity of

f

x=1

. Clearly state the type of discontinuity, if any, and justify your conclusion.