91Ó°ÊÓ

Continuity is a fundamental concept in calculus that describes the smooth behaviour of a function at a particular point or over an interval. A function is said to be continuous at a point \( a \) if the following conditions are satisfied:

• The function \( f(x) \) is defined at \( x = a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) is equal to the function value at \( a \), i.e., \( \lim_{x \to a} f(x) = f(a) \).

In the exercise, the function \( f(x) = |x-2| \) is continuous at \( x = 2 \) because both the left-hand limit and the right-hand limit as \( x \) approaches 2 are equal to \( f(2) = 0 \). This aligns with our understanding: for a function to be continuous, you can draw its graph without lifting your pencil. This continuity at \( x = 2 \) is verified by showing that \( \lim_{x \to 2^{-}} f(x) = \lim_{x \to 2^{+}} f(x) = 0 \).

Differentiability is about the ability to compute a tangent at any given point on a graph of a function. A function \( f(x) \) is said to be differentiable at a point \( a \) if the limit defining the derivative exists at that point. Specifically, differentiability at \( a \) requires that both the left-hand and right-hand derivatives exist and are equal:

• The left-hand derivative at \( a \): \( f_{-}^{\prime}(a) = \lim_{h \to 0^{-}} \frac{f(a+h) - f(a)}{h} \)
• The right-hand derivative at \( a \): \( f_{+}^{\prime}(a) = \lim_{h \to 0^{+}} \frac{f(a+h) - f(a)}{h} \)

For \( f(x) = |x - 2| \), these derivatives are \( -1 \) from the left and \( 1 \) from the right at \( x = 2 \). Since \( f_{-}^{\prime}(2) eq f_{+}^{\prime}(2) \), the function is not differentiable at this point. This discrepancy occurs because the graph has a sharp corner at \( x = 2 \).

Understanding limits is crucial for grasping the concepts of continuity and differentiability. The limit of a function \( f(x) \) as \( x \) approaches a point \( a \) is the value that \( f(x) \) gets closer to as \( x \) moves near \( a \). Limits help to define instantaneous rates of change and slope of tangent lines. They are defined as:

• The limit of \( f(x) \) as \( x \to a^- \) (from the left): \( \lim_{x \to a^-} f(x) \).
• The limit of \( f(x) \) as \( x \to a^+ \) (from the right): \( \lim_{x \to a^+} f(x) \).

In practice, for the function \( f(x) = |x - 2| \) around \( x = 2 \), both the left and the right limits approach \( 0 \). This confirms the function is continuous there, but their importance becomes even more prominent when analyzing the differentiability. Limits allow us to detect points of discontinuity or where a function is not smooth, such as cusps or sharp corners shown in the graph of \( |x - 2| \). Knowing how to compute these limits is key for evaluating the behavior of functions as inputs change.