![]() In the beginning, Sal drew an arbitrary continuous function over a closed interval to visually confirm that a continuous function does indeed have a maximum point and minimum point inside a closed interval. The same cannot be said for functions that do not satisfy these conditions, although it is possible to find or construct such functions that have a maximum and minimum over a closed interval. The extreme value theorem states that a function that is continuous over a closed interval is guaranteed to have a maximum or minimum value over a closed interval. ![]() Hope this helps you understand it a li'l bit better! ) The steps are neither continuous nor differentiable, because you obviously can't draw them without lifting your pen, and they have breaks in them, so you can't find the slopes at every point. The heart and star may be continuous, but they are not differentiable, because they have a pointy edge at which you cannot calculate the slope. However, you can't find the slope if the graph is shaped funny, like a heart or a star, or happens to have a jump in it, like a set of steps. I'm not sure how far along you are in calculus, but as you progress, you'll learn to find the slope of a curve, just like how you learned to find the slope of a line. ![]() So that function moves into this elite class of functions that we call continuous and differentiable continuous because you can draw it without stopping and differentiable because you can take the derivative at every point on the curve. A, you know that function is something you can draw without lifting your pen, and b, you know that function's smooth and doesn't have any kinks or pointy edges in it. Euler deserves the credit for a considerable proportion of modern mathematical notation.The fact that a graph is continuous makes a big difference when doing calculus. Similar remarks hold concerning Leibniz's differential signs as against Newton's signs for fluxions and infinitesimal increments. Newton's notation does not directly offer such possibilities. As a result, the notation $\int y\, dx$ is also suited for writing formulas for transformation of variables and is readily used for multiple and line integrals. Leibniz's notation $\int y\, dx$, while hinting at the actual process of constructing an integral sum, also includes explicit indication of the integrand and the variable of integration. It is worth emphasizing the essential advantage of Leibniz' integral symbol over Newton's proposal, namely the incorporation of the $x$. In particular, it was he who invented the modern differentials $dx, d^2 x, d^3 x$ and the integral The creator of the modern notation for the differential and integral calculus was G. Wallis (1655) had proposed the symbol $\infty$ for infinity. \varsigma'
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