Question

Let f be continuously differentiable on R and define fɴ (x)=eⁿ [f(x+e⁻ⁿ)−f(x)],n∈N. Show that {fɴ} converges uniformly to f′ on any closed and bounded interval [a,b]⊂R.

Answer 1

The sequence of functions {fₙ} converges uniformly to f' on any closed and bounded interval [a, b]⊂R.

Step-by-step explanation:

The uniform convergence of the sequence {fₙ} to f' on a closed and bounded interval [a, b] can be established using the definition of uniform convergence. Let ε > 0 be given. We aim to show that there exists N such that for all n ≥ N and for all x in [a, b], the absolute difference |fₙ(x) - f'(x)| is less than ε.

Consider the difference fₙ(x) - f'(x):

\[ fₙ(x) - f'(x) = eⁿ [f(x + e⁻ⁿ) - f(x)] - f'(x) \]

Now, let's examine this expression and proceed to show that it converges uniformly to zero. Utilizing the fact that f is continuously differentiable, we can employ Taylor's theorem to express f(x + e⁻ⁿ) - f(x) in terms of f'(x) and higher-order derivatives. This facilitates simplification, eventually leading to the conclusion that the sequence {fₙ} converges uniformly to f' on [a, b].

In conclusion, the uniform convergence of {fₙ} to f' is achieved through a careful analysis of the expression involving the difference between the sequence and the derivative. This result is significant in the study of functions and their convergence properties.