The following statements are false. Provide a counterexampleto support the claim
ID: 2940107 • Letter: T
Question
The following statements are false. Provide a counterexampleto support the claim. d) If on a closed interval[a,b], a function is bounded, takes its maximum and minimum values,takes all its values between the maximum and minimum value then thefunction is continuous on [a,b].Explanation / Answer
a) Take a constant function; it takes the maximum=minimum valueinfinitely many times (in fact, at all points). b) Let fn(x)=xn in the interval [0,1].Clearly, fn(x) is continuous for all n but as n->, fn converge to a function that is 0 on[0,1) and is 1 at x=1 which is clearly not continuous. c)Consider f(x)=x on (0,1). It is differentiable but as xapproaches 0, f'(x)= 1/2x approaches . However, as xapproaches 0, f(x) approaches 0. d) Take the function f(x)= x-[x] on the interval [0,2) and letf(x)=1 at x=2. Clearly this function is not continuous (notcontinuous at x=1) but it satisfies all the other propertiesstated..