Question

Prove that if \(\lim _{\Delta x \rightarrow 0} f(c+\Delta x)=f(c),\) then \(f\) is continuous at \(c\)

Short answer

The function \(f\) is continuous at \(c\) because \(\lim_{\Delta x \rightarrow 0} f(c+\Delta x) = \lim_{x \rightarrow c} f(x) = f(c)\), which is the definition of continuity at a point.

Step-by-step solution

Understand the given

We are given that \(\lim_{\Delta x \rightarrow 0} f(c+\Delta x) = f(c)\). This means that the value of \(f(c+\Delta x)\) gets arbitrarily close to \(f(c)\) as \(\Delta x\) becomes arbitrarily small.

Apply the definition of continuity

A function \(f\) is said to be continuous at a point \(c\) if \(\lim_{x \rightarrow c} f(x) = f(c)\). This is equivalent to saying that the value of \(f(x)\) gets arbitrarily close to \(f(c)\) as \(x\) becomes arbitrarily close to \(c\). But, if we let \(x = c+ \Delta x\), then as \(\Delta x\) becomes arbitrarily small, \(x\) becomes arbitrarily close to \(c\). Thus, \(\lim_{x \rightarrow c} f(x) = \lim_{\Delta x \rightarrow 0} f(c+\Delta x)\).

Proof of continuity

Given that \(\lim_{\Delta x \rightarrow 0} f(c+\Delta x) = f(c)\), the condition for \(f\) to be continuous at \(c\) is satisfied. Hence, \(f\) is continuous at \(c\).

Key concepts

Limits in Calculus

In calculus, the concept of a limit is fundamental for understanding how functions behave near a certain point. A limit describes the value that a function approaches as the input (or the independent variable) gets very close to a certain number. Essentially, it helps us define the behavior of functions at points where they might not be explicitly defined or where their behavior is changing.

Consider the scenario where we're looking at the function value of f(c + \text{Δ}x) as \text{Δ}x approaches 0. The expression
\[\lim_{\text{Δ}x \to 0} f(c + \text{Δ}x)\]
represents what value the function f is getting closer to, as we make \text{Δ}x smaller and smaller, basically as we move closer to the point c. If this limit exists and is equal to f(c), then the function is smoothing out at c, rather than having a jump or break.

Definition of Continuity

The definition of continuity at a point is a formal way to express the intuitive idea that a function can be drawn without lifting your pen from the paper. Formally, a function f is continuous at a point c if the following three conditions are met:

• The function f(x) is defined at c, which means f(c) exists.
• The limit of f(x) as x approaches c exists.
• The limit of f(x) as x approaches c is equal to f(c).

The expression
\[\lim_{x \to c} f(x) = f(c)\]
is a succinct way to capture all three of these conditions into a single equation, symbolizing the smooth passing of the function through the point c.

Function Continuity at a Point

To discuss function continuity at a point, let's take a closer look at our initial exercise. To prove that a function f is continuous at a point c, we should show that as x approaches c, the value of f(x) gets arbitrarily close to f(c). This is evident in the exercise we have, which states
\[\lim_{\text{Δ}x \to 0} f(c + \text{Δ}x) = f(c)\]
Translation: as the change in x, denoted by Δx, gets very small, effectively meaning as x gets close to c, the function's value mirrors the value at the point c.

If this limit equals f(c), then by the definition of continuity, f is continuous at c. It's like confirming that there's no gap or interruption in the function's value as we hone in on c. Therefore, if the limit of f(x) as x approaches c equals the value of the function at c, then f glides through c without any jolts or leaps, which is precisely what continuity at a point means.