Additional homework (only PL): problems, solutions.

Below I define notion, which are used in the above problems, which have not appeared till now during the classes.

### Cyclometric functions

In the same way as the logarithm is the inverse function to the exponential function, we can define the inverse functions for the trigonometric functions, i.e. arcus sinus, arcus cosinus, arcus tangens:

- iff ,
- iff ,
- iff ,

### Derivative of the inverse function

If is strictly monotone and continuous, then if derivative exists in and is non-zero.

E.g. let . Then . Therefore:

### Derivatives and inequalities

Using derivatives one can prove also some inequalities. If are functions, and , and na , then on . Indeed, if at the beginning of the interval is below , and increases slower than , then it is below in the whole interval.

E.g. on , because , and on .

### Lipschitz’s condition

We shall say that a function satisfies Lipschitz’s condition on the interval , if there exists , such that for all , we have . We see that Lipschitz condition is stronger than continuity.

### Uniform continuity

If we are able to choose for each , in a such way that in any interval of length values of functions do not differ more than , universally regardless of place , then we say that the function if uniformly continuous. More formally a function is uniformly continuous, if:

Function is a very simple example. It is uniformly continuous, because for any set . Then for all such that , then obviously .

On the other hand, function on the interval is continuous but is not uniformly continuous, because if , regardless how small we choose, we can find , such that , and therefore .