Let f:R R be defined as f(x)=x^α sin1x^β for x≠ 0 and f(0)=0, where α ,β ∈R. Which of the following is true? ( R denotes the set of all real numbers)

A. f(x) is continuous at x=0 for all α >0 and β ∈R. Step 1: For x≠ 0, |f(x)|≤ |x|^α because |sin1x^β |≤ 1. Step 2: If α >0, |x|^α 0, so f(x) 0=f(0) whatever β is: (A) is true. Step 3: (B) fails: α =-1, β =1 gives 1/xsin1/x, unbounded near 0. (D) fails: α =β =1 gives f(h)/h=sin1/h, which has no limit. Check: β =0 gives f(x)=x^α sin1 0 for α >0, so continuity does not need β >0. Trap: (C) adds the condition β >0, which is not needed for continuity.

Question 14 of 14 · Continuity & Differentiability

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Let $f : R \to R$ be defined as $f (x) = x^{α} s in \frac{1}{x ^{β}}$ for $x \neq = 0$ and $f (0) = 0$ , where $α, β \in R$ . Which of the following is true? ( $R$ denotes the set of all real numbers)

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $f (x)$ is continuous at $x = 0$ for all $α > 0$ and $β \in R$

Detailed Solution

For $x \neq = 0$ , $∣ f (x) ∣ \leq ∣ x ∣^{α}$ because $s in \frac{1}{x ^{β}} \leq 1$ .

If $α > 0$ , $∣ x ∣^{α} \to 0$ , so $f (x) \to 0 = f (0)$ whatever $β$ is: (A) is true.

(B) fails: $α = - 1$ , $β = 1$ gives $\frac{1}{x} s in \frac{1}{x}$ , unbounded near 0. (D) fails: $α = β = 1$ gives $\frac{f ( h )}{h} = s in \frac{1}{h}$ , which has no limit.

✓ Check

β = 0

gives

f (x) = x^{α} s in 1 \to 0

for

α > 0

, so continuity does not need

β > 0

⚠ Common trap

β > 0

, which is not needed for continuity.

Quick tip

$∣ x^{α} sin (x^{- β}) ∣ \leq ∣ x ∣^{α} \to 0 for α > 0$

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Continuity & Differentiability — Preparation

This question tests your understanding of Continuity & Differentiability, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A function is continuous at a point if you can draw its graph through that point without lifting your pen. Formally, f(x) is continuous at x = a if the left-hand limit, the right-hand limit and the value f(a) all exist and are equal. A function is differentiable at a point if its derivative exists there, which means the left-hand and right-hand derivatives are equal. This topic covers checking continuity and differentiability at a point and over an interval, the types of discontinuity (removable and jump), and the algebra of continuous and differentiable functions. The most important link is that every differentiable function is continuous, but a continuous function need not be differentiable. Graphs with a sharp corner, such as |x| at x = 0, are continuous but not differentiable there. The most common exam question gives a piecewise function with an unknown constant k and asks for the value that makes it continuous or differentiable at a point. Find the limit or derivative from each side, set them equal, and solve for k. Polynomial, exponential, sine and cosine functions are continuous everywhere, so you only need to check the points where a piecewise definition changes or where a denominator becomes zero. Practise the continuity and differentiability questions below, each with a step-by-step solution.

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Let f:R→R be defined as f(x)=xαsinxβ1​ for x=0 and f(0)=0, where α,β∈R. Which of the following is true? ( R denotes the set of all real numbers)

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Continuity & Differentiability — Preparation

Let $f : R \to R$ be defined as $f (x) = x^{α} s in \frac{1}{x ^{β}}$ for $x \neq = 0$ and $f (0) = 0$ , where $α, β \in R$ . Which of the following is true? ( $R$ denotes the set of all real numbers)