1. ## continuity problems

I am having a little problem with proving that these functions are continuous.

1) f(x) = (x+1+|x-2|)/(|4-x|+2x);
It is clear that the function is defined for all x belonging to R - {-4}. But how do I prove that the function is continuous on this interval.

2) f(x) defined on the interval [0,1] by:
f(x) = x (if x belongs to Q)
f(x) = 1 - x (if x does not belong to Q)
Show then that f(x) is continuous in only one point, and that point is 1/2.

Thanks in advance for the help guys.

2. Originally Posted by tombrownington
I am having a little problem with proving that these functions are continuous.

1) f(x) = (x+1+|x-2|)/(|4-x|+2x);
It is clear that the function is defined for all x belonging to R - {-4}. But how do I prove that the function is continuous on this interval.
Its easy if you know to break the definition of f(x) piecewise...

A general idea is:

If x > 4, then

f(x) = (x+1+x-2)/(-(4-x)+2x) = (2x - 1)/(3x - 4)

If 2 < x <= 4, then

f(x) = (x+1+x-2)/(4-x+2x) = (2x - 1)/(x + 4)

If x < 2, then

f(x) = (x+1-(x-2))/(4-x+2x) = 3/(x + 4)

Now firstly, check for continuity at the breakpoints, i.e. 2,4...

Then secondly, justify why(or why not) the piecewise definitions are continuous.

3. Originally Posted by tombrownington
2) f(x) defined on the interval [0,1] by:
f(x) = x (if x belongs to Q)
f(x) = 1 - x (if x does not belong to Q)
Show then that f(x) is continuous in only one point, and that point is 1/2.
Regarding this problem if I equate the two definitions of f(x):
x = 1 - x , and then solve for x I get the solution 1/2. But what argument can I give for this?
Any pointers?