TIFR-2018-Maths-A: 25

Question

True/False Question :

Let $f(x)$ and $g(x)$ be uniformly continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Then their pointwise product $f(x)g(x)$ is uniformly continuous.

Answer 1

Functions: $f(x)=x$ and $g(x)=x$.

Property: Both are uniformly continuous (their slope is a constant $1$).

Product: $h(x)=x^2$.

The derivative of the product is $h'(x)=2x$.

Look at the derivative, $h'(x)=2x$. The value of this derivative depends on $x$. As $x \to \infty$, the slope $2x \to \infty$. This means the slope is unbounded. A function with an unbounded slope cannot be uniformly continuous.

FALSE STATEMENT  

 

 

Comments

This reference on Uniform Continuity can help for beginners to understand : https://en.wikipedia.org/wiki/Uniform_continuity