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

Let f : [0, π] → R be defined byf (x) =(0 if x = 0,x sin 1x − 1x cos 1x if x̸ = 0.Is f continuous?

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.

Let f and g be two twice differentiable functions defined form R → R. If f (x) has a local maximum at x = c and g(x) has a local minimum at x = c, then*

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 :

Let R be the set of real numbers and f : R → R be the function defined by f (x) = 4x + 5.Show that f is invertible and find f −1