91Ó°ÊÓ

When we talk about a function being continuous, it simply means that the function behaves nicely without any breaks or jumps at a certain point. A function is continuous at a point if you can draw it without lifting your pencil from the paper.
The more formal definition requires that, for a function \( f(x) \) to be continuous at \( x = c \), the limit of the function as \( x \) approaches \( c \) should be equal to the function's value at \( c \). This means:

• \( \lim_{x \to c} f(x) = f(c) \)

This concept ensures that there are no sudden or unexpected changes in the value of the function near \( c \). If there is a disruption where the limit and the function's value do not match, the function is said to be discontinuous at that point. Continuous functions are important in calculus and many real-world applications where stable and predictable behavior is required.

Limits help us understand the behavior of a function as it approaches a particular point or extends towards infinity.
They are fundamental in defining derivatives and integrals in calculus. A limit is essentially the value that a function \( f(x) \) approaches as \( x \) gets closer to some number \( c \). This can be expressed as:

• \( \lim_{x \to c} f(x) \)

This concept is crucial because it tells us about the function's intended behavior, even if it's not defined at \( c \).
For continuous functions, limits play a key role since, at continuity points, the limit of a function as it approaches a point is the function's value at that point.
This is vital when analyzing functions like \( \sqrt{f(x)} \), ensuring they smoothly satisfy the required conditions for continuity.

The square root function \( \sqrt{f(x)} \) is a special kind of function that all students encounter in mathematics. Unlike other functions, it only makes sense when dealing with non-negative numbers. This is because you can't take the square root of a negative number within the realm of real numbers.
For the function \( \sqrt{f(x)} \) to be continuous at a point \( x = c \), we need \( f(x) \geq 0 \) around that point. If \( f(x) \) fulfills this condition, the limit of the square root function becomes

• \( \lim_{x \to c} \sqrt{f(x)} = \sqrt{\lim_{x \to c} f(x)} \)

This must equal \( \sqrt{f(c)} \).
So even if \( f(x) \) is continuous, you must ensure the values it takes are non-negative to keep \( \sqrt{f(x)} \) continuous. It is a subtle but crucial point!

Non-negative functions are those where the output is always zero or more, never dipping into negative values. They are important in our exercise because the square root of a negative number is undefined in real numbers, hence for \( \sqrt{f(x)} \) to exist and be continuous, \( f(x) \) must be non-negative at least around the point of interest \( x = c \).
The requirement, \( f(x) \geq 0 \), ensures that \( \sqrt{f(x)} \) remains a real-valued function. A helpful condition for the problem analyzed: if \( f(x) \) flips to negative, even just approaching \( c \), it can break the continuity of \( \sqrt{f(x)} \). This makes non-negative functions crucial when considering the continuity of square root functions, maintaining all results within the bounds of real numbers.