Answer to Question #5022 in Real Analysis for Christine

Question

Suppose f is continuous on all real numbers and f(c) is not equal to zero for some point c in all real numbers. Show that there exists an interval (a, b) containing c such that f is never 0 on this interval

Answer 1

f(c) is not equal to zero. Let f(c)>0. Then there exists such epsilon>0:
f(c)>epsilon.
As f is continuous function in c, then for each
epsilon1>0 there exists delta>0 such that for each x:
if
abs(c-x)<delta, then abs(f(c)-f(x))<epsilon1.
Let epsilon1 = epsilon /
2. Then there exists delta1 such that:
abs(f(c)-f(x))<epsilon/2 for each x
from (c-delta1, c+delta1). As f(c)>epsilon, then f(x) > epsilon / 2 >
0.
Let a = c-delta1, b = c+delta1, (a,b) contains c. So, we get that for each
x from (a,b), f(x)>0.
The proof is the same for f(c)<0.