An arithmetic function is any function which is defined for all positive integers, and has values which are either real or complex.
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Yes, it is.
Trigonometric functions, exponential functions are two common examples.
Usually none. A function can be identified as f but it is more often denoted by f(x) to show that it is a function of x.
In statistics, a graph and a chart are the same. In arithmetic, a graph is the plot of a function over values. There are no charts.
Sure. There are other applications of arithmetic, but algebra without arithmetic is impossible. A broad knowledge of arithmetic is essential for mastery of algebra.