# Real Function/Examples/Arbitrary Function/Mistake

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## Source Work

1963: Morris Tenenbaum and Harry Pollard: *Ordinary Differential Equations*:

- Chapter $1$: Basic Concepts:
- Lesson $2 \text B$: The Meaning of the Term
*Function of One Independent Variable*

- Lesson $2 \text B$: The Meaning of the Term

## Mistake

*The relationship between two variables $x$ and $y$ is the following. If $x$ is between $0$ and $1$, $y$ is to equal $2$. If $x$ is between $2$ and $3$, $y$ is equal to $\sqrt x$. The equations which express the relationship between the two variables are, with the end points of the interval included,*

\(\text {(a)}: \quad\) | \(\ds y\) | \(=\) | \(\ds 2,\) | \(\ds 0 \le x \le 1,\) | ||||||||||

\(\ds y\) | \(=\) | \(\ds \sqrt x,\) | \(\ds 2 \le x \le 3.\) |

*These*two*equations now define $y$ as a function of $x$. For each value of $x$ in the specified intervals, a value of $y$ is determined uniquely. The graph of this function is shown in Fig. $2.211$. Note that these equations do not define $y$ as a function of $x$ for values of $x$ outside the two stated intervals.*

**Figure $2.211$**

## Correction

The shape of the second part of the graph is incorrect.

It has been depicted as a convex function, in shape more like a square function than a square root.

The square root is a concave function.

The actual shape of the function in question can be found in the depiction of this arbitrary function.

## Sources

- 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term*Function of One Independent Variable*: Example $2.21$