Answer to Question #5022 in Real Analysis for Christine
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
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.