Let f, g : R → R be continuous functions such that f (a)̸ = g(a) for somea ∈ R. Show that ∃ a δ > 0 such that f (x)̸ = g(x), ∀x such that |x − a| < δ

Let f, g : R → R be given functions. Suppose that f and g are continuous at c ∈ R.Prove that the functionl(x) := inf{f (x), g(x)}, x ∈ R,is continuous at c.

Problem 2. Let f : R → R be a function. Suppose that f is increasing: for any x, y ∈ R,if x ≤ y then f (x) ≤ f (y). Prove that limx→a− f (x) exists for any a ∈ R. (Hint: What shouldthe value of this limit be? Try sup{f (x) : x < a}.)

Problem 4. Define the function f : R → R by f (x) = max{0, x}. For each a ∈ R, determineif f is differentiable at a and prove your answer

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.

Problem 5. Let f : R → R be a function, and suppose that f is differentiable at a for anya ∈ R.(a) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x) + c isdifferentiable at any a ∈ R with g′(a) = f ′(a).(b) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x + c) isdifferentiable at any a ∈ R with g′(a) = f ′(a + c).1