Graph of Continuous Functions : Real space to Normed linear space
Graph of Continuous Functions : Real space to Normed linear space M. Velrajan We discuss how learning the closedness of the graph of a continuous real valued function on real space, by a way of asking questions, thinking mathematically and applying various concepts leads to an extension of it to metric spaces, topological spaces and normed linear spaces. Graph of Continuous Functions 1. Recall that to draw the graph of real functions y = f(x) first we find the values of y for some of the values of x, for example x = 0, 1, -1, 2, -2, then we plot the points (x, f(x)) for these values of x and then we join all these plotted points by a free hand curve. This is because the functions y = f(x) we consider are continuous functions. The graph (curve) of continuous functions has no break, that is why such functions are called continuous functions. By the by, we note that the graph of a real function is the collection of all points (x, f(x)) in the xy-plane, ℝ × ℝ...