Question

Discuss the continuity of the function f, where f is defined by

f(x) = 2x, if x < 0
0, if 0<=x<=1
4x, if x>1

Answer 1

A function f(x) is continuous at a point x = a iff

i). f(a) is defined,
ii). limit of f(x) exists as x "approaches" a and equals L
This is true iff\[\lim_{x \rightarrow a ^{-}}f(x)=\lim_{x \rightarrow a ^{+}}f(x)=L\]iii).\[\lim_{x \rightarrow a}f(x)=f(a)\]If any of these conditions are not met, then f(x) is discontinuous at x = a.

There are three types of discontinuities:

I). Removable Discontinuity:
i). f(a) may or may not be defined,
ii). limit of f(x) exists as x "approaches" a and equals L, whether f(a) is defined or
not. However if f(a) is defined, then limit of f(x) as x "approaches a equals L.

II). Infinite Discontinuity:
i). f(a) is NOT defined,
ii). limit of f(x) as x "approaches" a from the left or the right equals negative or positive infinity. In other words, the limit does not exist as x "approaches" a.
Note: vertical asymptotes are infinite discontinuities.

III). Jump Discontinuity:
i). f(a) may or may not be defined,
ii). limit of f(x) as x "approaches" DOES NOT EXIST.
However for this type of discontinuity, there arises a special case. If f(a) is defined, then iff
i).\[\lim_{x \rightarrow a ^{-}}f(x)=f(a)\]then the function f(x) is continuous from the left, or if
ii). \[\lim_{x \rightarrow a ^{+}}f(x)=f(a)\]then f(x) is continuous from the right.

Answer 2

CORRECTION: PLEASE NOTE THE FOLLOWING ERROR!!! FOR REMOVABLE DISCONTINUITIES, f(a) is NOT DEFINED but he LIMIT DOES EXIST!!!!