Question

\(\mathrm{T} / \mathrm{F}\) : If \(f\) is continuous at \(c,\) then \(\lim _{x \rightarrow c} f(x)\) exists.

Short answer

True: If \(f\) is continuous at \(c\), \(\lim _{x \rightarrow c} f(x)\) exists.

Step-by-step solution

Understand the Problem

We need to verify if the statement "If \(f\) is continuous at \(c\), then \(\lim _{x \rightarrow c} f(x)\) exists" is true or false. Continuity at a point \(c\) implies certain conditions for the function \(f\).

Define Continuity at a Point

A function \(f\) is continuous at a point \(c\) if \(\lim _{x \rightarrow c} f(x) = f(c)\). This means the function approaches the value \(f(c)\) as \(x\) approaches \(c\) from either side.

Analyze the Implication

If the function \(f\) is continuous at \(c\), it directly means that \(\lim _{x \rightarrow c} f(x) = f(c)\). By definition, for a limit to exist, the left-hand limit and the right-hand limit must both exist and be equal to each other and to \(f(c)\).

Conclude the Solution

Since the definition of continuity at \(c\) includes the existence and equality of \(\lim _{x \rightarrow c} f(x)\) to \(f(c)\), it follows that if \(f\) is continuous at \(c\), \(\lim _{x \rightarrow c} f(x)\) exists by definition.

Key concepts

Limit of a function

The concept of a "Limit of a function" is fundamental in calculus. When we talk about the limit of a function as it approaches a particular point, we're exploring the behavior of the function values as they get infinitely close to that point. This does not require the function to actually reach the point, just that it approaches it closely.
In mathematical terms, for a function \( f(x) \), the limit as \( x \) approaches \( c \), expressed as \( \lim_{x \to c} f(x) \), asks: "What value does \( f(x) \) get closer to, as \( x \) gets closer to \( c \)?"
For a limit to truly exist, the values of \( f(x) \) must get close enough to a single finite number as \( x \) nears \( c \) from both sides (left and right). This means:

• The left-hand limit: \( \lim_{x \to c^-} f(x) \)
• The right-hand limit: \( \lim_{x \to c^+} f(x) \)

The expression \( \lim_{x \to c} f(x) \) only exists when both of these side limits exist and are equal.

Point of continuity

A function is said to be continuous at a point, often referred to as the "Point of continuity," when the function behaves predictably around that point. This means there's no sudden jumps, breaks, or holes in the graph of the function.
For a function \( f \) to be continuous at a specific point \( c \), three specific conditions must be met:

• The function \( f(c) \) must be defined. There has to be a real number result when you input \( c \) into the function.
• The limit \( \lim_{x \to c} f(x) \) must exist. This implies a convergence of function values from both directions towards a common value as \( x \) approaches \( c \).
• The actual function value at \( c \) equals the limit value, which is \( f(c) = \lim_{x \to c} f(x) \).

When these conditions are satisfied, it tells us that moving towards point \( c \) from either side brings you smoothly to \( f(c) \), indicating no disruption in the function at that point.

Existence of limits

The "Existence of limits" is a core prior condition needed for many calculations in calculus. Simply put, defining \( \lim_{x \to c} f(x) \) means that as \( x \) gets closer to \( c \), the values \( f(x) \) approach a particular number from both sides.
Let's dive into what it takes for a limit to exist. We need:

• Both the left-hand limit (approaching from values smaller than \( c \)) and the right-hand limit (from values greater than \( c \)) must be found and must both be numbers.
• These numbers must be equal. If they differ, the limit does not exist.

When we state that a limit exists, we are essentially saying that the function behaves consistently around that point, with no directionality discrepancies (from the left or right). In terms of establishing continuity, this is an integral part because without an existent limit, continuity can't even be considered.