Understanding Continuity of the Function $f(x) = [2x]$

This solution explores the continuity of the function $f(x) = [2x]$, where $[y]$ denotes the greatest integer function (or floor function). The greatest integer function gives the largest integer less than or equal to $y$. We need to determine at which of the given points the function $f(x)$ is continuous.

Conditions for Function Continuity

A function $f(x)$ is considered continuous at a specific point $x=a$ if it meets the following three essential criteria:

1. Function Definition: The function must have a defined value at the point $a$. That is, $f(a)$ must exist.
2. Limit Existence: The limit of the function as $x$ approaches $a$ must exist. This requires the limit from the left side to be equal to the limit from the right side: $$ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) $$
3. Limit Value Equals Function Value: The calculated limit of the function as $x$ approaches $a$ must be exactly the same as the function's value at $a$: $$ \lim_{x \to a} f(x) = f(a) $$

Continuity Behavior of the Greatest Integer Function

The fundamental property of the greatest integer function, $g(y) = [y]$, is that it experiences jumps (discontinuities) precisely at integer values of $y$. For any value $y$ that is not an integer, the function $[y]$ is continuous.

In our specific function, $f(x) = [2x]$, the input to the greatest integer function is $2x$. Consequently, $f(x)$ will exhibit discontinuities whenever the expression $2x$ results in an integer. If we set $2x = n$, where $n$ represents any integer, the points of discontinuity for $f(x)$ are found at $x = n/2$. This means the function $f(x) = [2x]$ is discontinuous at points such as ..., $-1$, $-0.5$, $0$, $0.5$, $1$, $1.5$, $2$, $2.5$, and so on.

Analyzing Continuity at Specific Points for $f(x) = [2x]$

Let's carefully check the continuity of $f(x) = [2x]$ at each of the points given in the options.

Continuity Check at $x=1$

For the point $x=1$, we evaluate $2x = 2 \times 1 = 2$. Since $2$ is an integer, the greatest integer function $[2x]$ will be discontinuous at this point.

• The function value is $f(1) = [2 \times 1] = [2] = 2$.
• Left-hand limit: As $x$ gets closer to $1$ from values less than $1$ (e.g., $0.99$), $2x$ gets closer to $2$ from below (e.g., $1.98$). Thus, $\lim_{x \to 1^-} [2x] = \lim_{y \to 2^-} [y] = 1$.
• Right-hand limit: As $x$ gets closer to $1$ from values greater than $1$ (e.g., $1.01$), $2x$ gets closer to $2$ from above (e.g., $2.02$). Thus, $\lim_{x \to 1^+} [2x] = \lim_{y \to 2^+} [y] = 2$.

Because the left-hand limit ($1$) is not equal to the right-hand limit ($2$), the function $f(x) = [2x]$ is discontinuous at $x=1$.

Continuity Check at $x=0.5$

At the point $x=0.5$, the expression inside the greatest integer function is $2x = 2 \times 0.5 = 1$. As $1$ is an integer, the function is expected to be discontinuous.

• The function value is $f(0.5) = [2 \times 0.5] = [1] = 1$.
• Left-hand limit: As $x \to 0.5^-$ (e.g., $x=0.49$), $2x$ approaches $1$ from below (e.g., $0.98$). So, $\lim_{x \to 0.5^-} [2x] = \lim_{y \to 1^-} [y] = 0$.
• Right-hand limit: As $x \to 0.5^+$ (e.g., $x=0.51$), $2x$ approaches $1$ from above (e.g., $1.02$). So, $\lim_{x \to 0.5^+} [2x] = \lim_{y \to 1^+} [y] = 1$.

The left-hand limit ($0$) differs from the right-hand limit ($1$). Therefore, the function $f(x) = [2x]$ is discontinuous at $x=0.5$.

Continuity Check at $x=2.5$

For the point $x=2.5$, we calculate $2x = 2 \times 2.5 = 5$. Since $5$ is an integer, the function $f(x) = [2x]$ is discontinuous here.

• The function value is $f(2.5) = [2 \times 2.5] = [5] = 5$.
• Left-hand limit: As $x \to 2.5^-$ (e.g., $x=2.49$), $2x$ approaches $5$ from below (e.g., $4.98$). So, $\lim_{x \to 2.5^-} [2x] = \lim_{y \to 5^-} [y] = 4$.
• Right-hand limit: As $x \to 2.5^+$ (e.g., $x=2.51$), $2x$ approaches $5$ from above (e.g., $5.02$). So, $\lim_{x \to 2.5^+} [2x] = \lim_{y \to 5^+} [y] = 5$.

Since the left-hand limit ($4$) is not equal to the right-hand limit ($5$), the function $f(x) = [2x]$ is discontinuous at $x=2.5$.

Continuity Check at $x=1.25$

Let's consider the point $x=1.25$. Here, the value inside the greatest integer function is $2x = 2 \times 1.25 = 2.5$. Crucially, $2.5$ is not an integer. According to the properties of the greatest integer function, $f(x) = [2x]$ should be continuous at points where the argument ($2x$) is not an integer.

• 1. Function Definition: $f(1.25) = [2 \times 1.25] = [2.5] = 2$. The function is defined at $x=1.25$.
• 2. Limit Existence:
  • Left-hand limit: As $x$ approaches $1.25$ from the left (e.g., $x=1.24$), $2x$ approaches $2.5$ from values slightly less than $2.5$ (e.g., $2.48$). So, $\lim_{x \to 1.25^-} [2x] = \lim_{y \to 2.5^-} [y] = 2$.
  • Right-hand limit: As $x$ approaches $1.25$ from the right (e.g., $x=1.26$), $2x$ approaches $2.5$ from values slightly greater than $2.5$ (e.g., $2.52$). So, $\lim_{x \to 1.25^+} [2x] = \lim_{y \to 2.5^+} [y] = 2$.
  Since the left-hand limit ($2$) is equal to the right-hand limit ($2$), the limit exists: $\lim_{x \to 1.25} [2x] = 2$.
• 3. Limit Value Equals Function Value: We found that $\lim_{x \to 1.25} f(x) = 2$ and $f(1.25) = 2$. These values are equal, satisfying the third condition for continuity.

As all three conditions for continuity are met at $x=1.25$, the function $f(x) = [2x]$ is indeed continuous at this point.

Summary of Function Continuity Points

The function $f(x) = [2x]$ is discontinuous solely at points $x$ where $2x$ is an integer (i.e., where $x = n/2$ for any integer $n$). Conversely, it is continuous at all points where $2x$ is not an integer.

Our detailed checks confirmed:

• $f(x)$ is discontinuous at $x=1$ because $2x=2$ is an integer.
• $f(x)$ is discontinuous at $x=0.5$ because $2x=1$ is an integer.
• $f(x)$ is discontinuous at $x=2.5$ because $2x=5$ is an integer.
• $f(x)$ is continuous at $x=1.25$ because $2x=2.5$ is not an integer.

Therefore, the function $f(x) = [2x]$ is continuous at $x=1.25$.