Let f : R → R be defined byf (x) = 2023 : x ∈ Q,x : x /∈ Q.(a) Prove, directly from the definition of sequential continuity, that f issequentially continuous at x = 2023 and not sequentially continuous atany other point. [10 marks](b) At which points is f differentiable? Justify your answer with reference toappropriate theorems.

Consider the function f : R → R defined by f (x)=1 if x∈ Q, f (x)=0 if x∈ R/Q, where is f continuous? be sure to prove your assertion

Problem 5. Let g : R → R be a function and a ∈ R.(a) Prove that if g is differentiable at a, then there exists function φ : R → R that iscontinuous at a and satisfiesg(x) − g(a) = (x − a) · φ(x) for all x ∈ R.(Hint: We proved a very similar statement within the proof of the chain rule.)(b) Conversely, prove that if there exists function φ : R → R that is continuous at a andsatisfies the above equation, then g is differentiable at a.

(a) For which x-values does the function appear to be not continuous? (Select all that apply.)012345(b) For which x-values does the function appear to be not differentiable? (Select all that apply.)012345

Let f(x)=∣∣2x2+5∣∣x|−3|,x∈R. If m and n denote the number of points where f is not continuous and not differentiable respectively, then m+n is equal to :

You Let f : [0, 1] → R, where f(x) := (1 if x ∈ {1/n: n ∈ N},0 otherwiseNote that f is Riemann integrable on [0, 1] by a previous assignment. Let F : [0, 1] → R, F(x) := integral 0 to x (f).(a) Prove that F is differentiable on [0, 1].(b) Find a c ∈ [0, 1] such that f is discontinuous at c and F'(c) = f(c).(c) Find a d ∈ [0, 1] such that f is discontinuous at d and F'(d) 6= f(d).Remark: This shows that if the indefinite integral F of a Riemann integrable function f isdifferentiable at a point c where f is discontinuous, then F'(c) may or may not equal f(c).