Question

The function f(x) = {((π/4) + tan^-1 x, |x| ≤ 1), ((1/2)(|x| - 1), |x| > 1) is :

(1) continuous on R–{1} and differentiable on R – {–1, 1}. 

(2) both continuous and differentiable on R – {–1}. 

(3) continuous on R – {–1} and differentiable on R – {–1, 1}. 

(4) both continuous and differentiable on R –{1}

Answer 1

(1) continuous on R–{1} and differentiable on R – {–1, 1}.

For continuity at x = –1 

L.H.L. = (π/4) - (π/4) = 0

R.H.L. = 0 

So, continuous at x = –1 

For continuity at x = 1 

L.H.L. = 0 

R.H.L. = (π/4) + (π/4) = π/2 

So, not continuous at x = 1 

For differentiability at x = –1 

L.H.D. = 1/(1 + 1) = 1/2

R.H.D. = – 1/2 

So, non differentiable at x = –1