- #1

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- Homework Statement
- prove the continuity of the function

- Relevant Equations
- f,g

Let f be continuous in [0,1] and g be continuous in [1,2] and f(1)=g(1). prove that

$$

(f*g)=

\begin{cases}

f(t), 0\leq t\leq 1\\

g(t), 1\leq t \leq2

\end{cases}$$

is continuous using the universal property of quotient spaces.

Let ##f:[0,1]→X## and ##g:[1,2]→Y##

f and y are continuous, thus for open sets in [0,1] and open sets in [1,2], their images are also open in X and Y, hence X and Y are topological spaces.

f(1)∈X is an element in X and g(1)∈Y is an element in Y.

let W be a discrete union of X and Y, which itself is a topological space

##f(1)=g(1)## is an equivalence relation on W.

Then (f*g) is a projection map from W onto W/(f(1) ~g(1)) and hence continuous by the universal property of quotient spaces

$$

(f*g)=

\begin{cases}

f(t), 0\leq t\leq 1\\

g(t), 1\leq t \leq2

\end{cases}$$

is continuous using the universal property of quotient spaces.

Let ##f:[0,1]→X## and ##g:[1,2]→Y##

f and y are continuous, thus for open sets in [0,1] and open sets in [1,2], their images are also open in X and Y, hence X and Y are topological spaces.

f(1)∈X is an element in X and g(1)∈Y is an element in Y.

let W be a discrete union of X and Y, which itself is a topological space

##f(1)=g(1)## is an equivalence relation on W.

Then (f*g) is a projection map from W onto W/(f(1) ~g(1)) and hence continuous by the universal property of quotient spaces

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