91Ó°ÊÓ

In topology, a set is termed "closed" if its complementary set is open. To understand what this means, consider the set we're evaluating: the graph of the function \( \Gamma_f \). For our function \( f \), the graph is composed of two distinct parts: the hyperbola-like curve \( \{ (x, \frac{1}{x}) \mid x > 0 \} \), and the line segment \( \{ (x, 0) \mid x \leq 0 \} \) along the non-positive part of the x-axis.

The critical aspect of proving that this graph is a closed set involves looking at what happens to sequences of points within this set. If any sequence of points \( (x_n, f(x_n)) \) within the graph converges to a point \( (a, f(a)) \) in \( \mathbb{R}^2 \), that limit point must also lie on the graph.

• When \( x_n \gt 0 \), the function is \( f(x_n) = \frac{1}{x_n} \), and as \( x_n \to a \gt 0 \), then \( f(x_n) \to \frac{1}{a} \).
• If \( x_n \leq 0 \), the function is \( f(x_n) = 0 \).

These convergences demonstrate that both limit points are part of the graph, affirming \( \Gamma_f \) as a closed set.

Finding the complement (everything not in \( \Gamma_f \)) that remains open in the plane hence helps prove the graph is closed.

Continuity of a function involves the behavior of the function as it approaches a point from all directions. For a function \( f \) to be continuous at a point \( a \), the limit as \( x \) approaches \( a \) must equal \( f(a) \). In simpler terms, \( \lim_{x \to a} f(x) = f(a) \). For our function, let's focus particularly at the point \( x = 0 \). Here, the function is defined as \( f(x) = 0 \) for \( x \leq 0 \). However, notice that approaching zero from the right (i.e., \( x \to 0^+ \)), the function \( f(x) = \frac{1}{x} \) tends towards infinity.This behavior tells us something crucial:

• Approaching from the right results in \( \lim_{x \to 0^+} f(x) = \infty \).
• Approaching from the left (\( x \leq 0 \)), \( \lim_{x \to 0^-} f(x) = 0 \).
• Since \( f(0) = 0 \), the limits do not match: \( \lim_{x \to 0} f(x) eq f(0) \).

This discrepancy indicates a discontinuity at \( x = 0 \), and thus, \( f \) is not continuous at this point.

The graph of a function is a visual representation of the set of all ordered pairs \( (x, f(x)) \). For the given function \( f \), the graph is partitioned into two segments:

• A hyperbolic curve \( \{ (x, \frac{1}{x}) \mid x > 0 \} \) in the first quadrant, which corresponds to parts of the curve where \( x \) is positive.
• The x-axis for \( \{ (x, 0) \mid x \leq 0 \} \), representing all negative and zero values where the function constantly outputs zero.

Understanding these parts helps in visualizing how the function behaves on different domains.

When faced with such graphs, one should pay attention to:

• The domain restrictions, which provide context for each section of the graph.
• The possible asymptotic behavior, like the approach towards \( x = 0 \) from both directions.
• The junction points where different forms of the function meet, as these often indicate points of interest related to continuity and limit behavior.

By examining such features, the behavior of the function can be predicted and understood even without explicit calculations, offering a powerful tool in mathematical analysis.