Name: Rain Xia       Student Number: 59118159

In a continuous function, there is an interval [a,b] in x-axis, the corresponding y value of a and b is f(a) and f(b). There has a point c in x-axis which is bigger than a and smaller than b, and f(c) is between the values of f(a) and f(b) definitely. It is called Intermediate Value Theorem.

Like when a<0 and b>0, there must has a point c which equals to 0 if a<c<b. It has the same idea of the previous one.

Because [a,b] is a closed interval under the continuous function, [f(a),f(b)] is still a closed interval. Therefore we can find a value we want in the interval definitely.

My argument is relying on the concept of continuity. The Intermediate Value Theorem must in a continuous function, the function is continuous in [a,b]. If the function is not continuity, the Intermediate Value Theorem is not true anymore.

Name: Rain Xia       Student Number: 59118159

The example I made for function is fountain. When you look at the picture at this fountain, you will find out that the track of the water looks like a quadratic function, which is represented as y=ax²+bx+c(a≠0). I guess the function for the half of the fountain is may be y=-x²-5x+10 if we make the ground as the x-axis and fountain center as the 0 point. And this function does not have a horizontal asymptote.

The example I made for sequence is the transverse wave of particle. Many wave phenomena in nature can be represented by the sine mathematical function. This sequence converges to 0.

The example I made for series is a person want to walk 1 km, he walks 0.5km first time and he walks half of the remaining distance every time after. The sum of his each walking is a series. This series does not converge but arbitrarily close to 1.