Question

Discuss the continuity of the following function at the indicated point:

`f(x) = {{:(|x| cos (1/x)",", x ≠ 0),(0",", x = 0):} at  x = 0`

Answer 1

Given:

`f(x) = {{:(|x| cos (1/x)",", x ≠ 0),(0",", x = 0):} at  x = 0`

Consider,

`lim_(x->0) f(x) = lim_(x->0)|x| cos(1/x)`

= `lim_(x->0) f(x) = lim_(x->0) |x| lim_(x->0) cos (1/x)`

= `lim_(x->0) f(x) = 0 xx lim_(x->0) cos (1/x) = 0`

= `lim_(x->0) f(x) = f(0)`

Hence, f(x) is continuous at x = 0