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The concept of continuity is fundamental to understanding how functions behave. In simple terms, a function is continuous at a point if there is no 'break' or 'jump' in the function at that point. What this means is that the function's value at the point and the limit of the function as it approaches that point from either direction are the same. For functions to be continuous over an interval, they must be continuous at every point within that interval.

Polynomial functions, like the function given in the exercise, are a classic example of continuous functions. They do not have any breaks, jumps, or points of discontinuity – you can draw them without lifting your pencil from the paper. This property of polynomials is extremely useful when applying concepts like the Intermediate Value Theorem, as it guarantees the function will not 'jump' over any values between any two points on the graph.

Polynomial functions are algebraic expressions that involve only non-negative integer powers of the variable. For example, the function in the exercise, f(x) = x^3 + x + 1, is a third-degree polynomial, because the highest exponent on x is three. Polynomials are particularly nice to work with because they have a number of well-understood properties:

• They are continuous and differentiable everywhere, which means they have no sudden changes in direction or breaks.
• The degree of the polynomial tells us the maximum number of zeros or roots the function can have.
• They are made up of 'simpler' functions (power functions) which combine to form the overall behavior of the polynomial.

Understanding these properties allows us to analyze the behavior of polynomial functions and predict where the function will cross the x-axis, which corresponds to finding the zeros of the function.

Finding the zeros of a function involves locating the x-values at which the function equals zero. In other words, it's where the curve of the function crosses the x-axis on a graph. Zeros are important because they provide critical information about a function’s behavior and are used in solving equations.

The Intermediate Value Theorem, or IVT, comes in handy especially with continuous functions like polynomials. The IVT states that if a function is continuous on a closed interval [a, b] and takes on different signs at those endpoints, then it must cross over zero at least once within that interval. The theorem essentially guarantees the existence of at least one zero without having to find its exact value.

In our example exercise, we demonstrated this concept by calculating the values of f(x) at x = -1 and x = 0, finding that the function changed from negative to positive, thereby confirming via the IVT that there must indeed be a zero between these two points.