Are you familiar with the chain rule? You will need to use that here.

For So,Let be the real-valued function defined by

(a) Find

When So, remembering that sine is an odd function ( ), we have(b) Find

Why would you think that? Sine is continuous over the entire real line.(c) Determine whether is continuous at . Justify your answer

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For c I said no because sin is never continuous at 0?

is everywhere continuous, is everywhere continuous, and is everywhere continuous. The composition of two continuous functions is itself continuous, so and are both continuous. Finally, the sum of two continuous functions is continuous, so is everywhere continuous (including ).

If you like, you could also use the definition of continuity at a point by showing that (to do this, consider the one-sided limits).

See above. is continuous at and in fact, is also differentiable at Consider the left- and right-sided limits.(d) Determine whether the derivative of exists at . Justify your answer.

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Finally for d i out that it does not because is not continuous at 0.

and

Both one-sided limits are equal, so the limit exists.