Question

Which of the following function is continuous but not uniformly continuous on the real line?

• A. \(\sin x\)
• B. \(x^2\)
• C. constant function
• D. Identity function

Hint:

Polynomial \(x^2\) is continuous on \(\mathbb{R}\), but not uniformly continuous on \(\mathbb{R}\).

Answer 1

The Correct Option is B

Concept:
A function can be continuous on the whole real line but still fail to be uniformly continuous. Uniform continuity requires that the same \(\delta\) works for all points in the domain.

Step 1: Check \(x^2\). The function \[ f(x)=x^2 \] is continuous on \[ \mathbb{R} \] because it is a polynomial.

Step 2: Why \(x^2\) is not uniformly continuous on \(\mathbb{R}\). For two close points \(x\) and \(y\), \[ |x^2-y^2|=|x-y||x+y| \] Even if \[ |x-y| \] is very small, the factor \[ |x+y| \] can become very large when \(x,y\) are large. So one fixed \(\delta\) cannot control the difference for all real numbers.

Step 3: Check other options. \[ \sin x \] is uniformly continuous on \(\mathbb{R}\). A constant function is uniformly continuous. The identity function \(f(x)=x\) is also uniformly continuous.

Step 4: Final answer. \[ \boxed{x^2} \]