1/1-x not uniformly continuous on [0,1)

Question

show that the continuous function f(x)= 1/1-x is not uniformly continuous on [0,1).

_1--

Answer 1

Here is an intuitive way to look at it:

 

If x approaches 1 from the left, 1/(1 - x) approaches +∞.  But if x approaches 1 from the right, 1/(1 - x) approaches -∞.  You can see there is an infinite discontinuity at x = 1.

Answer 2

Although not a formal proof, consider f(0.98) and f(0.99) as a counterexample.

 

f(0.98) = 1/(1-0.98) = 1/0.02 = (2/100)^(-1) = 50

while

f(0.99) = 100

 

So it goes from 50 to 100 in 0.01, a rate of change of 5000 for a tangent line slope differential

 

Therefore, there exists x and y in [0,1) where the function values do NOT stay inside the unit rectangle.

 

This violates the definition of uniform continuity