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Linear functions represent one of the most fundamental types of functions in algebra. They have the form of f(x) = mx + b, where m is the slope, indicating how steep the line is, and b is the y-intercept, which tells us where the line crosses the y-axis. Their graph is a straight line, hence the term 'linear'.

One key characteristic of linear functions is that they are continuous and do not have breaks, holes, or jumps in their graphs. They can be defined for all real numbers, which comprises their domain. It is the simplicity of this form that grants the predictability and continuous nature of linear functions. As long as the values of x are within the domain— which, in the case of pure linear functions, is all real numbers ( R )— the function will provide a single output for every input without any interruption or discontinuity.

In calculus, limits are used to describe the behavior of a function as the input approaches a certain value. They are essential for understanding concepts like continuity, derivatives, and integrals. Formally, the limit of f(x) as x approaches c is the value that f(x) gets closer to as x gets arbitrarily close to c.

Limits can be used to check for continuity at a point by verifying if the left-hand limit, right-hand limit, and the value of the function at the point in question are all equal. If they are, the function is continuous at that point. However, in the case of linear functions addressed in the previous section, limits are not necessary to establish continuity due to their consistent nature, which inherently lacks abrupt changes or boundaries.

The domain of a function is the complete set of possible values of the independent variable, or inputs, for which the function is defined. It essentially answers the question, 'For what values of x does this function give a real number output?' The domain is critical when examining function properties such as continuity.

When you are given a function, to determine its domain, you look for values of x that might cause the function to become undefined. This might include dividing by zero, taking the square root of a negative number, or finding the logarithm of a non-positive number in real number context. Linear functions have the simplest domain: all real numbers ( R ), because there are no restrictions on the input values; they can take any real number and produce a real number output, reaffirming their inexhaustible continuity.