Q: Let {xmsin⁡1x;x≠00;x=0 Find the set of values of m for which (i) f(x) is continuous at x=0 (ii) f(x) is differentiable at x=0 (iii) f(x) is continuous but not differentiable at x=0.

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Q: Let {xmsin⁡1x;x≠00;x=0  Find the set of values of m for which   (i) f(x) is continuous at x=0   (ii) f(x) is differentiable at x=0  (iii) f(x) is continuous but not differentiable at x=0.

Bernard Lacey · Accepted Answer

The given function is:  {xmsin⁡1x;x≠00;x=0  i) f is continuous at x=0 if   f(0)=limx→0f(x)  Now, limx→0f(x)  =limx→0xmsin⁡1x  x=0 is m&amp;gt;0  limx→0f(0)≠0 if m&amp;lt;0  δ0 f is continuous  for mt(0,∞)

Jenny Bolton · Accepted Answer

ii) Now,&amp;nbsp;f′(x)=m⋅xm−1⋅sin⁡1x+xm⋅cos⁡(1x)⋅(−1x2)mxm−1⋅sin⁡(1x)−xm−2cos⁡(1x)&amp;nbsp;if&amp;nbsp;x≠0So, it is differentiable at&amp;nbsp;x=0if&amp;nbsp;m&amp;gt;2&amp;nbsp;because then only&amp;nbsp;f(x) is differentiable&amp;nbsp;Hence, f is continuous but only differentiable at&amp;nbsp;x=0&amp;nbsp;if&amp;nbsp;m∈[2,∞)

karton · Accepted Answer

iii) So, f is continuous but only differentiable at (0,∞);[2,∞) is on (0, 2)

Q: Let {(x^{m}\sin\frac{1}{x};,x \ne 0),(0;,x=0):} Find the set of values of

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