Question

Continuity of a Function Discuss the continuity of the function \(h(x)=x[x]\) .

Short answer

The function \(h(x) = x[x]\) is discontinuous at all integer values of \(x\).

Step-by-step solution

Handling integer part

The function's behavior for integer values of \(x\) needs to be analyzed first. It's important to remember that for any integer \(n\), [n] = n. So if \(x = n\), \(h(n) = n * n = n^2\). Hence, for integer values of \(x\), the function simplifies to a square, which is always continuous.

Handling non-integer part

For non-integer values, take an arbitrary non-integer \(x = n+a\), where \(n\) is an integer part and \(a\) is a fractional part (0 < a < 1). In this case, our function becomes \(h(x) = (n+a) * n\), because [.] gives the largest integer less than or equal to \(x\). Recognizing this is continuous in the domain (n,n+1), because the function here is a linear function multiplied by an integer, which is also continuous.

Determining continuity

By examining the limits of the function for integer and fractional parts separately, we see that in each case the function is continuous. However, it's key to check if the function is same at adjacent integer and fractional boundaries. If \(n\) is any integer, the right limit at \(n\) (i.e., \(lim_{x->n^{+}}h(x)\)) is \(n*n = n^2\), and the left limit at \(n\) (i.e., \(lim_{x->n^{-}}h(x)\)) is \((n-0)* (n-1) = n(n-1)\). As they are unequal, the function \(h(x) = x[x]\) is discontinuous at all integer values.

Key concepts

Integer Part

The integer part of a number is a fundamental concept in understanding certain kinds of mathematical functions. For a real number, its integer part (often denoted as [x]) is essentially the largest integer less than or equal to the number.
For example, if you have a number like 3.7, its integer part is 3.
Now, looking at the function \( h(x) = x[x] \), when \( x \) is an integer, like 3, the integer part is 3 itself, and the function evaluates to \( 3 \times 3 = 9 \).
The behavior of the function at integer values is straightforward and consistent because the integer part does not introduce any additional complexity.
Thus, for integer values of \( x \), the function is simply a square of the integer, leading to it being smooth and continuous.

Non-integer Values

Non-integer values introduce a slight twist to functions like \( h(x) = x[x] \). When \( x \) isn't a whole number, you can express it as \( x = n + a \), where \( n \) is an integer and \( a \) is the fractional part of \( x \) such that \( 0 < a < 1 \).

For these values, the function can be rewritten as \( h(x) = (n + a) \cdot n \).
Here, the floor function (integer part) always returns the integer \( n \), allowing the function to behave like a linear one within the open interval \( (n, n+1) \). This linearity ensures continuity because no abrupt changes occur as \( x \) changes within the interval between two consecutive integers.

• Linear functions are inherently continuous within their domain.

As a result, the function \( h(x) = x[x] \) remains continuous for non-integer \( x \), provided \( x \) does not exactly hit an integer.

Limits of the Function

The continuity of a function often requires examining the limits at specific points.
For the function \( h(x) = x[x] \), evaluating limits as \( x \) approaches an integer reveals its quirks. Consider an integer \( n \).

The right-hand limit, as \( x \) approaches \( n \) from the right (notated as \( \lim_{x \to n^+} h(x) \)), equals \( n^2 \).
Similarly, the left-hand limit, as \( x \) approaches \( n \) from the left (notated as \( \lim_{x \to n^-} h(x) \)), differs as it results in \( n(n-1) \).
Since these two limits don't match, the function cannot be continuous at integer points.

• Continuous functions require equal left and right limits.

Therefore, although \( h(x) \) manages to be continuous for non-integers, the differing limits at integer values indicate a break in continuity.

Discontinuity at Integer Points

Discontinuities are points where a function doesn't "flow" perfectly. For \( h(x) = x[x] \), these discontinuities occur at each integer point.
Even though the function behaves continuously between integers, reaching an integer value brings about inconsistency. When \( x \) is exactly an integer, the value of the function sharply jumps.

• This is due to the mismatch in limits we've explored earlier.

For example, the function value as we approach from the right and left side of an integer \( n \) does not match the function's exact value at \( n \).
Such points of discontinuity are common in functions involving the integer part, as the definition of [.] inherently causes sudden changes whenever you cross an integer threshold.
As a result, the function \( h(x) = x[x] \) is discontinuous at integer values, clearly highlighting why understanding limits is crucial.