#### MathematicalPhysicist

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M(t)=sup{f(x)|a<=x<=t} prove that M(t) is a continuous function on [a,b].

2) let f be a continuous function on [a,b], and we define that A={x in [a,b]| f(x)>=0} where f(a)>0>f(b).

i need to show that the supremum of A, s, that f(s)=0.

3) i need to prove/disprove that f(x)=arctan(x) is a uniform continuous on R.

for the first question, i think bacuase f is continuous then by wierstrauss theorem f achieves its max and min in the interval, so also M(t) is bounded, and the supremum of f(x) in a<=x<=t is also achieved in this interval and thus M(t) is continuous, i think this is basically the idea but i need to formalise this, anyone can help me on this?

and the second one, i showed that A is bounded and non-empty so it has got a supremum, now we have two options, or s in A or s isnt in A.

if s is in A, then f(s)>=0, so in oreder to show that f(s)=0, i tried to show ad absurdum that f(s)>0 leads to contradiction, but i didnt succeded in it, where did i go wrong?

also any help on the third question will helpful.