Chapter 11.1, Problem 45E

To determine

To determine: Whether the statement is true or false.

In a parameterization, if x is a continuous function of t and y is a continuous function of t , then y is a continuous function of x .

Answer to Problem 45E

The given statement is false.

Explanation of Solution

Calculation:

It is known that in a parameterization, if x and y are two continuous functions of t ; then the functions can be written as:

  $x=g\left(t\right)$  .....(1)

  $y=h\left(t\right)$  ..... (2)

Here, g and h are continuous functions.

Rewrite the equation (1) as:

  $\begin{array}{c}x=g\left(t\right)\\ t=g^{-1}\left(x\right)\end{array}$

Substitute $g^{-1}\left(x\right)$ for t in the equation (2).

  $y=h\left(g^{-1}\left(x\right)\right)$

The functions $g^{-1}$ and $h\left(g^{-1}\right)$ are continuous functions as the function g is also continuous.

And, $h\left(g^{-1}\right)$ is a composite function of two continuous functions.

This means that y is not a continuous of x .

Hence, the given statement is false.