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Piecewise functions are a type of function made up of multiple sub-functions, each of which applies to a specific interval or piece of the domain. This means that instead of one formula defining the function across its entire domain, a piecewise function has different formulas for different sections of its domain. In our example, the function is defined as:

• \( f(x) = \frac{1}{1-x} \) for \( 0 \leq x < 1 \)
• \( f(x) = 0 \) when \( x = 1 \)

This means that for any \( x \) less than 1, we use the first formula, while exactly at \( x = 1 \), we use the second formula where \( f(x) \) is 0. Piecewise functions are helpful in modeling situations where a rule changes at a boundary, which is common in real-world problems.

In calculus, continuity of a function at a point means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous on an interval, it must be continuous at every point within that interval. In our function, \( f(x) \) is continuous on the open interval \([0, 1)\). It is not continuous at \( x = 1 \). This is because the value of the function jumps from what it would be as it approaches 1 from the left, to 0 exactly at 1.
To check continuity mathematically, we see if the limit of the function as \( x \) approaches a point equals the value of the function at that point. Since as \( x \to 1^- \), \( f(x) \to +\infty \), but \( f(1) = 0 \), the function is not continuous at \( x = 1 \). This discontinuity at the boundary cumulatively affects the closed interval \([0, 1]\), making it non-continuous.

Limits are fundamental in calculus as they help describe the behavior of functions as they approach specific points. The limit tells us what value a function is approaching as the input gets closer to a certain point, but not necessarily what the value is at that point. For \( f(x) \), the interesting limit is as \( x \) approaches 1 from the left, written as \( \lim_{{x \to 1^-}} f(x) \).
In the example, this limit is +\infty, meaning as \( x \) gets closer and closer to 1, \( f(x) \) grows larger without bound. Still, this is different from the value of the function at 1, which is defined to be 0. So while the limit gives us an idea of the behavior surrounding the point, it does not necessarily equate to the function's value at that point, showcasing a critical point about limits and continuity.

Extrema refer to the maximum and minimum values of a function. Every continuous function defined on a closed interval should have a minimum and a maximum value due to the Extreme Value Theorem. However, our function only satisfies part of this condition.

• The minimum value of \( f(x) \) on the interval \([0,1]\) is 0, occurring at \( x = 1 \).
• There is no maximum value because as \( x \to 1^- \), \( f(x) \to +\infty \). The function increases without bound as \( x \) approaches 1 from the left.

Therefore, the function has a lower bound but no upper bound within this domain. This happens because the function is not actually attaining \(+\infty\), instead it just perpetually increases. Understanding extrema in piecewise functions can thus be a bit more complex compared to straightforward continuous functions on closed intervals.