Let \(\Gamma\) be a Jordan are joining \(a\) to \(b\) in the complex plane \(\mathbb{C}\) and let \(f\) be a function continuous on \(\Gamma\). Assume that \(\int_\Gamma fdz\) is defined via Riemann sums. The author is concerned with the following questions (1) is the formula \(\int_\Gamma f(z)dz=F(b)-F(a)\), \(F\) being a primitive for \(f\) ``along \(\Gamma\)'', valid? (2) is the Rolle's theorem valid for functions on \(\Gamma\)? (3) Does the inequality \(| f'(z)|\leq M\), \(z \in\Gamma\) imply the condition \(| f(z_1)-f(z_2) |\leq L(z_1-z_2)\) for any \(z_1,z_2\) on \(\Gamma\)? The author gave conditions that guarantee affirmative answers to the above questions.