Continuity of Functions

Solved Problems

Solution.

Let $$. We must find some number $$ so that for all $$ with

the following inequality is valid:

We can write the last expression as

Hence,

Notice that our function takes only non-negative values. Therefore the $$-neighborhood at the given point must satisfy the condition $$. In this case the left side $$ of the inequality will be negative. It follows from here that

Thus, if we take $$, then for all $$ with $$, we will have $$. For example, if $$, then the value of $$ must be $$ $$ $$ This means, by Cauchy definition, that

Solution.

Let $$. Calculate the values of the function at $$ and $$

Hence, we have $$ and $$, or

By the intermediate value theorem, this means that there exists a number $$ in the interval $$ such that $$

Thus, the equation has a solution in the interval $$

Solution.

As the function $$ is a polynomial, it is continuous. We notice that

Therefore $$. Then we can conclude by the intermediate value theorem, that there exists a number $$ in the interval $$ such that $$. Thus, the equation has a solution in the interval $$

Solution.

The left-side limit at $$ is

Then the value of $$ at $$ must be equal to $$

Similarly, the right-side limit at $$ is

As you can see, the value of $$ at $$ must be equal to $$

For given values of $$ and $$, the function $$ is continuous. The graph of the function is sketched in Figure $$

Solution.

We calculate the left-hand and right-hand limits at $$

The function will be continuous at $$ if

Hence,