Let f:[a,b]-->R be continuous at c in [a,b] and suppose that f(c)>0. Prove there exist a positive number m and an interval [u,v] subset of [a,b] such that c in [u,v] and f(x)>=m for all x in [u,v]
Let f:[a,b]-->R be continuous at c in [a,b] and suppose that f(c)>0. Prove there exist a positive number m and an interval [u,v] subset of [a,b] such that c in [u,v] and f(x)>=m for all x in [u,v]
Take $\displaystyle m = \tfrac12f(c)$. Use the definition of continuity to show that $\displaystyle |f(x) - f(c)|<m$ for all x in some neighbourhood [u,v] of c. Then use the triangle inequality to show that f(x)>m for all x in [u,v].