Is g(x) Always Continuous if Sandwiched Between Two Continuous Functions?

[Cacophony]

Homework Statement

If f(x) < g(x) < h(x) for all x E R, and if f and h are continuous functions, must g also be continuous? If so, why? If not, can you come up with a counter example?

What do you think?

 

[blazeatron]

Hi...
For the given conditions ,the function g need not be continuous.
Example : f(x) = sin(x) - 3 , h(x) = sin(x) + 3 , g(x) = sin(1/x)
for this example , f(x) < g(x) < h(x) . f and h are continuous . But g is not continuous at x tends to 0. Instead of g(x) = sin(1/x) , u can also take a signum function which is not continuous at x = 0.

 

[HallsofIvy]

Blazeatron's example is sufficient but simpler-
f(x)= 0 for all x, h(x)= 3 for all x, g(x)= 1 if x is rational, 2 if x is irrational. g is discontinuous for all x.