Question

Find the points of discontinuity of the function f, where `f(x) = {{:(sinx",",  0 ≤ x ≤ pi/4),(cos x",", pi/4 < x < pi/2):}`

Answer 1

Clearly f(x) is defined at all points of `[0, pi/2]`

Case (i) Let x₀ ∈ `[0, pi/4]`

`lim_(x -> x_0) f(x) =  lim_(x -> x_0) sin x`

= sin x₀

`f(x_0)` = sin x₀

∴ `lim_(x -> x_0) f(x) = f(x_0)`

Hence f(x) is continuous at x = x₀.

Since x₀ is an arbitrary point of `[0, pi/4]`

f(x) is continuous at all poin of `[0, pi/4]`

Case (ii) Let x₀ ∈ `[pi/4, pi/2]` 

`lim_(x -> x_0) f(x) =  lim_(x -> x_0) cos x`

= cos x₀

`f(x_0)` = cos x₀ 

∴ `lim_(x -> x_0) f(x) = f(x_0)`

Hence, f(x) is continuous at x = x₀. 

Since x₀ is an arbitrary point of `[pi/4, pi/2]`

f(x) is continuous at all points of `[pi/4, pi/2]`

Hence, f (x) is continuous at all points `[0, pi/2]`.