Question

Examine the continuity of f, where f is defined by:

f(x) = `{(sin x - cos x", if"  x != 0),(-1", if"  x = 0):}`

Answer 1

f(x) = `{(sin x - cos x", if"  x != 0),(-1", if"  x = 0):}`

Approach 1:

If f(x) is continuous at x = c, it implies:

f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)

⇒ −1 = sin 0 − cos 0 = −sin 0 − cos 0

⇒ −1 = −1 = −1

This shows that f(x) is continuous at x = 0.

Approach 2:

c ≠ 0 and c ⊂ R

If f(x) is continuous at x = c, it implies:

sin c − cos c is continuous, i.e. sin c and cos c are continuous functions, which is true.

That is, f(x) is also continuous at x ≠ 0.