91Ó°ÊÓ

Continuity is a crucial concept in calculus that describes whether a function behaves "nicely" at a particular point. A function is continuous at a point if three conditions are met:

• The function is defined at that point, meaning there is a finite value, say \( f(c) \).
• The limit of the function as it approaches the point from both sides exists \( \lim_{x \to c} f(x) \).
• The value of the function at the point equals the limit of the function as it approaches the point \( f(c) = \lim_{x \to c} f(x) \).

Simply put, a function is continuous at a certain point if you can draw the function without lifting your pencil. In the given problem, even though \( f(1) = 5 \) is known, we can't assert continuity without knowing how the function behaves as \( x \) approaches 1.

The left-hand limit is the value that a function approaches as the input nears a particular point from the left-hand side. Mathematically, it is expressed as \( \lim_{x \to c^-} f(x) \).

Imagine approaching your destination only from one direction - the left side. The left-hand limit helps us understand the behavior of the function from this one particular direction.

In our exercise, for the limit \( \lim_{x \to 1} f(x) \) to exist, the left-hand limit \( \lim_{x \to 1^-} f(x) \) must coincide with the right-hand limit. Understanding both left and right limits is crucial as they provide detailed insights into the function's behavior near the point.

The right-hand limit is analogous to the left-hand limit but measures the behavior of the function as you approach a point from the right side. It is denoted as \( \lim_{x \to c^+} f(x) \).

Think of it as arriving from the right hand side of your destination. To ascertain that a limit exists at a certain point, both the right-hand limit and left-hand limit must exist and, importantly, be equal.

In the given calculus problem, the existence and the equality of the right-hand limit are essential. This helps in determining if the overall limit \( \lim_{x \to 1} f(x) \) exists. Without this information, just knowing \( f(1)=5 \) does not inform us about the behavior or limit as \( x \to 1 \).

Discontinuity occurs when a function fails to be continuous at a given point. This means at least one of the conditions for continuity is not met. Discontinuities can appear in different forms, such as "jump," "infinite," or "removable" discontinuities.

• Jump discontinuity: occurs when a function has distinct left-hand and right-hand limits.
• Infinite discontinuity: arises when the function goes towards infinity from either direction as it approaches a point.
• Removable discontinuity: is present when a function's limit exists but does not equal the function's actual value at that point.

In an exercise scenario like ours, if \( \lim_{x \to 1^-} f(x) eq \lim_{x \to 1^+} f(x) \), a discontinuity is present at \( x = 1 \). This can result in the limit not existing or being distinct from \( f(1) \). Understanding discontinuities is essential for graphically and analytically assessing complex functions.