SOLUTION: what do you consider that a function is continuous.How to see if it exists. Show by means of an example that limit as x approaches a[f(x)+g(x)] may exist even

Algebra ->  Functions -> SOLUTION: what do you consider that a function is continuous.How to see if it exists. Show by means of an example that limit as x approaches a[f(x)+g(x)] may exist even      Log On


   

Question 1145907: what do you consider that a function is continuous.How to see if it exists.

Show by means of an example that limit as x approaches a[f(x)+g(x)] may exist even through neither limit as x approaches a f(x) nor limit as x approaches a g(x)exists.
Show by means of example that limit as x approaches a [f(x)*g(x)] may exist even through neither limit as x approaches a f(x) nor limit as x approaches a g(x) exists
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.

Consider this function, defined on the number line for all real numbers

              / 0 if x is irrational,
             |  
    f(x) =  <
             | 
              \ 1 if x is rational.


This function is discontinued at every point on the number line.


Next consider function  g(x) = 1 - f(x).


This function is discontinued at every point on the number line, too.


But the sum of these two functions  f(x) +g(x) = f(x) + (1 - f(x)) == 1  is identically equal to 1 
in all number line and is, therefore, continue function in all points of the number line.