*Determine whether the following functions are defined and where they are continuous.*

, .

### Solution

As is defined everywhere, is defined as long as the denominator, . Now

,

therefore . So is defined for all except . As is continuous (as the sum of a polynomial and a sine) and is continuous (as a polynomial), is continuous as long as .

Similarly is defined for, and continuous at, all .

*By evaluating and , show that there is a solution to the equation .*

### Solution

and . As is continuous on the interval , satisfies the hypothesis of the Intermediate Value Theorem on :

*As is continuous, it takes all values between *

*and**— this is the Intermediate Value Theorem*Now, looking at the equation:

*,*

for which is certainly the case here in . Now so by the Intermediate Value Theorem, there exists a such that * — *which is equivalent to having a solution.

*Evaluate and , but show that there is no (real) solution to the equation lying in the interval . What is different to your analysis of ?*

and .

Let us find *all *solutions to .

,

which is fine as long as (if , is undefined so certainly not equal to .)

.

That is there are *no* real solutions to , in particular in the interval .

Although and , suggesting a such that , does not in fact satisfy the hypothesis of the Intermediate Value Theorem on as it is not continuous at and thus not continuous on .

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