Definition of convergence for functions between metric spaces. Uniqueness of the limit and characterization in terms of convergent sequences. Limits of composite functions. The case of complex functions: relationship with the limits of real part and imaginary part functions, limit of the sum, product and ratio of two functions. Limits involving the point at infinity. Functions continuous at a point, continuous functions. A function is continuous if and only if the inverse function transforms open sets into open sets or closed sets into closed sets. The composition of continuous functions is continuous. The case of complex functions: relationship with the continuity of real part and imaginary part functions, continuity of the sum, product and ratio of two continuous functions. Uniformly continuous and Lipschitz continuous functions: mutual implications. A continuous function transforms compact sets into compact sets and connected sets into connected sets. A continuous function on a compact set with values in \(\mathbb {R}\) has absolute maximum and minimum. The modulus of a continuous function on a compact set with values in \(\mathbb {C}\) has absolute maximum and minimum. A function with values in \(\mathbb {C}\) which is continuous and non-zero at a point is non-zero in a neighborhood of that point. A continuous function on a compact set is uniformly continuous.