(iii) The trigonometric functions are continuous on. To see this note that
Thus, we can choose for any given to satisfy the given requirement of continuity.
(iv) Let For any given , let be chosen such that . Then, for implies that .
Hence, is continuous at . For , given implies that
Hence, . Finally, for any using limit theorems, we have .
Hence, is continuous everywhere on .
(v) The function , is not continuous at c, if c is an integer since the left hand limit is not equal to the
right hand limit.
(vi) Consider the function if x is rational, and if x is irrational. It is discontinuous at every . To see this note that given , we can choose and both conveying to c, where each is a rational an each is an irrational. Then, . Thus, does not exist.
are continuous on
. To see this note that 
for any given 
to satisfy the given requirement of continuity. 
For any given 
, let 
be chosen such that
. Then, for 
implies that 
. 
is continuous at 
. For 
, given 
implies that 
. Finally, for any 
using limit theorems, we have 
. 
is continuous everywhere on 
.
, is not continuous at c, if c is an integer since the left hand limit is not equal to the
if x is rational, and 
if x is irrational. It is discontinuous at every
. To see this note that given 
and 
both conveying to c, where each 
is a rational an each 
is an irrational. Then, 
. Thus, 
does not exist. 
