Thomae function f(x):(0,1)->R f(x)=p/q x is rational number (p and q are relatively prime natural number) f(x)=0 x in irrational number
Thomae's function f(x) is a mathematical function that evaluates to 0 if x is irrational and to 1/n if x is a rational number expressed in lowest terms p/q, where n is the denominator of the fraction.
Proving that Thomae's function f(x) is not differentiable at any point is important because it is a counterexample to the common misconception that continuous functions are always differentiable. It also demonstrates the importance of carefully defining mathematical concepts and the limitations of certain mathematical concepts and tools.
To prove that Thomae's function f(x) is not differentiable, one can use the definition of differentiability, which requires the existence of a unique tangent line at each point on the function's graph. By showing that the function has sharp corners and no unique tangent line at rational points, it can be proven that the function is not differentiable.
The discontinuity of Thomae's function f(x) at rational points is significant because it shows that the function is not continuous and therefore not differentiable at those points. This challenges the commonly held belief that continuous functions are always differentiable.
While Thomae's function f(x) may not have any direct real-world applications, it is a valuable mathematical concept that helps to illustrate the limitations of certain mathematical tools and concepts. It also serves as a counterexample to common misconceptions about continuous functions and differentiability.