9. (a) A function f : R → R is said to be periodic on R if there exists a number t > 0 such tha f(x+t) = f(x) for all z € R. Prove that a continuous periodic function on R is bounded and uniformly continuous on R. (b) Is f(x) = sinx+cos uniformly continuous on R?

Definethefunctionf :R→Rbyf(x) x2 x⩾1. Showthatf iscontinuousat a = 1.

Let f(x)=x²+sin x+1 on the interval [1, 2]. Choose the correct statement below. O Since f is not continuous on [1,2], it does not attain its absolute maximum value nor its absolute minimum value. O By the Extreme Value Theorem, f attains both its absolute maximum value and its absolute minimum values. O Since f is not differentiable on [1, 2], it does not attain its absolute maximum value nor its absolute minimum value. The function f attains only its absolute minimum value. O The function f attains only its absolute maximum value.

2. LEFT AND RIGHT LIMITS (a) Prove that f: (0,1)→ R defined by f(x) = is not uniformly continuous. (b) Let f: R\ {0} → R be the function defined by f(x) = sin(). Prove that the left and right limits of f at 0 do not exist. (c) Let f: R\ {0} → R be the function defined by f(x) = 2√√1+. Find the left and right limits of f at 0.

3. Consider the function f(x) : (0, ∞) → R where f(x) = =. (1) Show by definition that f is continuous at every > 0. (2) Show that f does not have a right limit at 0.