Pointwise convergence

Treadstone 71

If a sequence of functions f_n converges pointwise to a bounded function f, does f have the intermediate value property? If not, are there some conditions that will make it so?

Euclid

No. Consider f_n = f for all f and f(x) = 0 for x<1 and 1 for x geq 1.

The intermediate value property holds for continuous functions. But even then pointwise convergence is not enough. Consider f_n=x^n on [0,1]. The limiting function is f(x)=0 for x in [0,1) and f(1)=1.

However, the limit of a sequence of continuous functions will be continuous if you have uniform convergence.

matt grime

Quote by Treadstone 71

If a sequence of functions f_n converges pointwise to a bounded function f, does f have the intermediate value property? If not, are there some conditions that will make it so?

As stated this is trivially false. Surely you mean f_n to be continuous at the very least.

Treadstone 71

I don't have uniform convergence. But suppose (f_n) is an increasing sequence of continuous functions on R. Suppose (f_n) converges pointwise to f, then {x in R : f(x)>a} must be open, and I can show this if I had the intermediate value property.

Treadstone 71

Actually I don't think this is true without uniform convergence. I can think of functions that satisfy the intermediate value property and yet {x in R : f(x)>a} is not open, even closed. I'll repost in the homework section.

HallsofIvy

Let f_n(x)= 0 is x<0, nx if 0<= x<= 1/n, 1 if x> 1/n.

Each such f is continuous but the sequence converges pointwise to
f(x)= 0 if x<= 0, 1 if x> 0 which is not continuous and does not satisfy the intermediate value property.

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Treadstone 71	Pointwise convergence Tweet If a sequence of functions f_n converges pointwise to a bounded function f, does f have the intermediate value property? If not, are there some conditions that will make it so?

Euclid	No. Consider f_n = f for all f and f(x) = 0 for x<1 and 1 for x geq 1. The intermediate value property holds for continuous functions. But even then pointwise convergence is not enough. Consider f_n=x^n on [0,1]. The limiting function is f(x)=0 for x in [0,1) and f(1)=1. However, the limit of a sequence of continuous functions will be continuous if you have uniform convergence.

Treadstone 71	Pointwise convergence I don't have uniform convergence. But suppose (f_n) is an increasing sequence of continuous functions on R. Suppose (f_n) converges pointwise to f, then {x in R : f(x)>a} must be open, and I can show this if I had the intermediate value property.

HallsofIvy Recognitions: Gold Member Science Advisor Staff Emeritus	Let f_n(x)= 0 is x<0, nx if 0<= x<= 1/n, 1 if x> 1/n. Each such f is continuous but the sequence converges pointwise to f(x)= 0 if x<= 0, 1 if x> 0 which is not continuous and does not satisfy the intermediate value property.