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Continuous functions are a fundamental concept in mathematics. They provide a way to describe smooth and unbroken graphs. A function is considered to be continuous if it is uninterrupted at every point in its domain. Imagine drawing a graph without lifting your pencil from the paper; that's akin to a continuous function.
To determine if a function is continuous at a point, three main criteria must be satisfied:

• The function must be defined at that point.
• The limit of the function as it approaches the point from both directions must exist.
• The value of the function at that point must be equal to the limit.

This means that small changes in the input should result in small changes in the output, without any jumps or gaps. For the function given in our exercise, checking these traits allows us to confirm its continuity.

Real numbers are the set of numbers that include both rational and irrational numbers. This means they cover every possible number line value that you can locate on a continuous line. They include fractions, integers, and numbers that have no exact fractional representation, like \(\pi\) and \(\sqrt{2}\).
In the context of our function, real numbers are essential because they make up the domain where we check continuity. For functions defined on real numbers, we often need to see whether there are inputs that may pose problems like making the function's denominator equal to zero. However, in our example, the expression \(x^2 + 4\) ensures there's no risk of division by zero, as it is always positive for any real \(x\). Thus, our function remains stable across the real number set.

Discontinuity in a function occurs where there are breaks, jumps, or undefined points in its graph. It typically means the function cannot be smoothly or continuously graphed at those points:

• One common source of discontinuity is division by zero, leading to undefined values.
• Vertical asymptotes, where a function tends to infinity, are another.
• Sometimes, piecewise functions, defined separately over intervals, can also pose problems at the junctions.

In our exercise, we checked for potential discontinuity by inspecting where the denominator becomes zero. Since \(x^2 + 4\) is always positive, the function \(f(x)=\frac{x+2}{x^2+4}\) possesses no real roots that might lead to discontinuity. This characteristic allows us to conclude confidently that the function is fully continuous over the set of real numbers.