Let f : R → R be continuous at c ∈ R. Then |f | is continuous at c. Give anexample to show that the reverse is not true

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

If f(x)=|x|,x∈R, then

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. Prove that the function f : R → R, f (x) = |x|3 is twice differentiable at anypoint a ∈ R, but is not three-times differentiable at 0.

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