What is wrong with the argument claiming to prove the following assertion. You should not only find the error, but explain why it is an error providing an example. Assertion: For any two numbers and define max(, ) to be the largest of or . If f and g are continuous at z then so is the function max(f, g) defined by max(f, g)(x) = max(f(x), g(x)). Proof: According to Definition 2.4.1 it must be shown that limxz max(f, g)(x) = max(f, g)(z). According to Definition 2.2.1, in order to show this, it must be shown that for every > 0 there is some > 0 such that | max(f, g)(x) max(f, g)(z)| < whenever 0 < |z x| < . So let > 0 be arbitrary. By the definition of max(f, g)(z) = max(f(z), g(z)) it must be the case that either max(f, g)(z) = f(z) or max(f, g)(z) = g(z).Only the first case will be considered since the argument in the second case is identical. If max(f, g)(z) = f(z) (1) then there must be some non-empty interval (z p, z + p) such that max(f, g)(x) = f(x) (2) for all x such that z p < x < z +p. Since f is continuous at z it follows by Definition 2.4.1 that limxz f(x) = f(z) and by Definition 2.2.1 that there is some > 0 such that |f(x) f(z)| < (3) whenever 0 < |x z| < . Now let be the minimum of and p. Then if 0 < |x z| < it follows that z p < x < z + p and so from (1) that max(f, g)(x) = f(x). But also 0 < |x z| < and so it follows from (3) that |f(x) f(z)| < . Combining (1), (2) and (3) yields that | max(f, g)(x) max(f, g)(z)| < as required