Continuity

Step 1: Continuity for $x%5Cneq%200$ .
For $x%5Cneq%200$ , $f(x)%3Dx%5Csin(1%2Fx)$ is a product of continuous functions, hence continuous.

Step 2: Limit at $x%3D0$ .
Since $-1%5Cle%20%5Csin(1%2Fx)%5Cle%201$ , multiplying by $x$ gives $-%7Cx%7C%5Cle%20x%5Csin(1%2Fx)%5Cle%20%7Cx%7C$ . As $x%5Cto%200$ , both bounds $%5Cto%200$ ; by the Squeeze Theorem, $%5Clim%5Climits_%7Bx%5Cto%200%7Dx%5Csin(1%2Fx)%3D0$ .

Step 3: Compare with $f(0)$ .
$f(0)%3D0$ , which equals the limit, so $f$ is continuous at $0$ .

Step 4: Conclusion.
$f$ is continuous for all $x%5Cin%5Cmathbb%7BR%7D$ .

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Continuity

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs