We give examples of sequences of functions and define the pointwise convergence of a sequence of functions in Sect. 7.1. In this chapter we will mainly investigate when the pointwise limit of a sequence of functions is continuous, differentiable, or integrable. As for continuity, the uniform convergence, defined in Sect. 7.2, will be shown to work well. We also discuss uniformly Cauchy sequences of functions in Sect. 7.2. And in Sect. 7.3, we will prove that the uniform limit of a sequence of continuous functions is continuous, and thus integrable. However, differentiability of a uniform limit does not work well under uniform convergence. Some uniformly convergent sequences of differentiable functions may not converge to a differentiable function. Nevertheless, by restricting our attention to power series, we will prove that a power series can be differentiated term-wise in Sect. 7.4, and finally this enables us to prove that \(\frac{d}{dx}e^{x} = e^{x}\) . Then we conclude this chapter by introducing Taylor’s theorem in Sect. 7.5.