Problem 35
Question
If \(f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)+f(0)}{3} \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).
Step-by-Step Solution
Verified Answer
The function \(f(x)\) is continuous for all x.
1Step 1: Assumption and substitution
Start with \(x = y = 0\) in the initial equation. This simplifies to \(f(0) = \frac{3f(0)}{3}\) which gives us \(f(0) = f(0)\), a tautology. No new information extracted here.
2Step 2: Construct new equation by substituting appropriate values
Now, put \(y = -x\) in the equation which gives us \(f(0) = \frac{f(x) + f(-x) + f(0)}{3}\). Rearranging, this simplifies to \(f(x) + f(-x) = 2f(0)\). Thus, for any given x, the sum of f(x) and f(-x) is constant.
3Step 3: Consider special case and algebraic rearrangement
Let us take a special case where \(y = -x\) and derive the expression again. This shows us that \(f(x) = f(-x)\) for all x, which suggests that the function \(f(x)\) is even.
4Step 4: Prove the continuity
We are given that the function is continuous at \(x = 0\). As per definition of continuity, this implies that \(\lim_{x\to 0} f(x) = f(0)\). Since we have found that \(f(x)\) is an even function, we can conclude that \(\lim_{x\to a} f(x) = f(a)\) for all a, or in other words, \(f(x)\) is continuous for all x.
Key Concepts
Even FunctionLimit of a FunctionFunction Continuity Proof
Even Function
The concept of an even function plays a crucial role in understanding the overall behavior of a mathematical function in terms of its symmetry. An even function is defined by the condition that for every number x in the function's domain, the value of the function at -x is the same as its value at x, hence the equation is expressed as \[ f(-x) = f(x) \].
This property indicates a symmetrical pattern about the y-axis when graphed on a Cartesian plane. If you imagine folding the graph along the y-axis, both halves would match up perfectly. This becomes particularly useful because it can simplify the understanding and computation of the function over its entire domain. In the context of our exercise, by setting y to -x and manipulating the original equation, we reach the conclusion that the function \(f(x)\) is even. This does not only confirm symmetry, but also is a stepping stone towards proving the continuity of the function for all x, not just at the single point where x equals zero.
This property indicates a symmetrical pattern about the y-axis when graphed on a Cartesian plane. If you imagine folding the graph along the y-axis, both halves would match up perfectly. This becomes particularly useful because it can simplify the understanding and computation of the function over its entire domain. In the context of our exercise, by setting y to -x and manipulating the original equation, we reach the conclusion that the function \(f(x)\) is even. This does not only confirm symmetry, but also is a stepping stone towards proving the continuity of the function for all x, not just at the single point where x equals zero.
Limit of a Function
The concept of the limit of a function is foundational in calculus and analysis, describing the behavior of a function as its input approaches a particular value. The formal definition specifies that a function \(f(x)\) has a limit \(L\) at a point \(a\) if, irrespective of how \(x\) approaches \(a\) (from left or right), the function’s value can get arbitrarily close to \(L\). Mathematically, this is expressed as \[\lim_{x\to a} f(x) = L\].
However, it's critical not to confuse a function having a limit at a point with the function actually being equal to that limit at the point. It's the approach that matters, not the actual arrival. This becomes a major player when discussing continuity, as continuity at a point requires that a function not only has a limit there but also that the function's value at that point is exactly equal to the limit.
However, it's critical not to confuse a function having a limit at a point with the function actually being equal to that limit at the point. It's the approach that matters, not the actual arrival. This becomes a major player when discussing continuity, as continuity at a point requires that a function not only has a limit there but also that the function's value at that point is exactly equal to the limit.
Function Continuity Proof
Proving a function's continuity at a point involves showing that three conditions are satisfied: the function is defined at the point, the limit of the function as it approaches the point exists, and the function's value at that point is equal to this limit. In our case, by definition, \(f(x)\) is continuous at \(x=0\) and, as shown in the steps, adheres to the equation \[\lim_{x\to 0} f(x) = f(0)\].
This exercise goes a step further, using the properties of even functions and the specified condition of the function to prove that \(f(x)\) is indeed continuous at any arbitrary point \(a\). It leverages the fact that continuity at zero and the function being even can be extended to infer continuity everywhere on its domain. This is a significant conclusion because it simplifies the analysis of the function and ensures predictable behavior for all inputs, a crucial aspect when applying mathematical models to real-world problems.
This exercise goes a step further, using the properties of even functions and the specified condition of the function to prove that \(f(x)\) is indeed continuous at any arbitrary point \(a\). It leverages the fact that continuity at zero and the function being even can be extended to infer continuity everywhere on its domain. This is a significant conclusion because it simplifies the analysis of the function and ensures predictable behavior for all inputs, a crucial aspect when applying mathematical models to real-world problems.
Other exercises in this chapter
Problem 33
If \(f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).
View solution Problem 34
If \(f(x+2 y)=f(x)+2 f(y)-2 f(0) \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).
View solution Problem 36
If \(f\left(\frac{x+2 y}{3}\right)=\frac{f(x)+2 f(y)}{3} \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).
View solution Problem 37
Show that a polynomial function \(P_{n}(x)=a_{n} x^{n}+a_{s-1} x^{n-1}+\ldots \ldots+a_{1} x+a_{0}\) is a continuous function.
View solution