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- The Heaviside function (named after the English mathematician Oliver Heaviside (1850 to 1925) who also has the ionic layer mentioned in the lyrics of the musical
*Cats*called after him) is defined by:

*h*(*x*) = 0 for*x*≤ 0 and*h*(*x*) = 1 for*x*> 0.

Prove that*h*is not continuous at 0 using:

(i) the sequential definition of continuity,

(ii) the*ε*-*δ*definition of continuity. - Use the
*ε*-*δ*definition of continuity to prove that the function*f*(*x*) =*x*^{2}is continuous at the point*x*= 1. - The function the
*floor*of*x*is defined to be*x*rounded down to an integer and is written*x*.

So for example the floor of 2.6 is 2 and the floor of -3.3 is -4 while the floor of 5 is 5.

Prove that this function is discontinuous at every integer and continuous elsewhere.

(You can probably guess what the*ceiling*of*x*is.)

The integer part function is denoted by [*x*]. So for example [-3.5] = -3 and [2.8] = 2.

Prove that this function is continuous at 0 but discontinuous at all other integers.

The fractional part of*x*is denoted by {*x*}. Where is this function continuous? - Let
*f*and*g*be functions which are continuous on the whole of**R**and with*f*(0) =*g*(0).

Prove that the function defined by*h*(*x*) =*f*(*x*) for*x*≤ 0 and*h*(*x*) =*g*(*x*) for*x*> 0 is continuous everywhere.

Hence prove that the absolute value function |*x*| is continuous everywhere.

If*f*is a continuous function on**R**, prove that the function |*f*(*x*) | is also continuous on**R**.

If*f*and*g*are continuous functions prove that the function*M*(*x*) = max{*f*(*x*),*g*(*x*)} is also continuous.

[Hint: Prove that max(*a*,*b*) =((*a*+*b*) + |*a*-*b*|)/2 for any real numbers*a*and*b*.]

Prove that the function*m*(*x*) = min{*f*(*x*),*g*(*x*)} is continuous if*f*and*g*are.

- Use this picture of the unit circle with the angle
*ABC*=*x*and angle*ABD*=*y*to prove that | sin*x*- sin*y*| ≤ |*x*-*y*|

Hence prove that the function*f*(*x*) = sin*x*is continuous everywhere.

Prove that the function cos*x*is continuous everywhere and hence locate the points where the function tan*x*is continuous and prove it.

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