Discuss the continuity on 0 ≤ x ≤ 1 & differentiability at x = 0 for the function.
f(x) = 
where x ≠ 0, x ≠ 
& f(0) = f (1/r π ) = 0, r = 1, 2, 3,.......
Text Solution
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continuous in 0 ≤ x ≤ 1 & not differentiable at x = 0
Sol. y = f(x) = xsin 1/x. sin 
when x ≠ 0, 
, r = 1,2,3
y = 0, x = 0, 
where r = 1,2,3,...............
Let t = x sin1/x as x → 0 + , t 
and as x 
, t 
0
y = t sin1/t
= f(0) also 
= f 
f(x) is continous at x = 0 and 
⇒ f(x) is continous 
We know that t = xsin1/x is not differentiable at x = 0
therefore y = tsin1/t = xsin1/x. sin 
is not differentiable at x = 0
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