Principles of Mathematics in Operations Research

11.4 Continuity and Connectedness 163

Fig. 11.2. Example 11.4.3

(i) If f(x+) or f(x—) does not exist, we say the discontinuity at x is of the
second kind,
(ii) If f(x+) and f(x—) both exist, we say the discontinuity at x is of the first
kind or simple discontinuity.
(Hi) If f(x+) = f(x-), but f is discontinuous at x, then the discontinuity at
x is said to be removable.

y=l-x

Fig. 11.3. Example 11.4.5

Example 11.4.5

/:K->K, f(x) = x, x e Q

l - x, x eM\i

/ is continuous (only) at x = |:

Let e > 0 be given. Let 5 = e. Let teRB\t — x\<6 where x—\.

t£Q=> \f(t) - f(x)\ = \t-x\ = \t-±\<8 = e.

ten\Q=>\f(t)-f(x)\ = \i-t-x.

Hence, f is continuous at x — -

_ (^11) |1 — x\ < 6 = e.
2'
CLAIM:f is discontinuous every other point than x —\
(without loss of generality, we may assume that x > ^):
Let x ^ . Show f(x+) does not exist. Let e = ' x 2 ~~ '. Assume f(x+) exists,