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प्रश्न
Examine the continuity of f, where f is defined by:
f(x) = `{(sin x - cos x", if" x != 0),(-1", if" x = 0):}`
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उत्तर
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.
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