Answer
See explanations.
Work Step by Step
Step 1. If the function $f(x)$ has no zeros in the interval of $(a,b)$, the function will not cross the x-axis, which means that the function will have the same sign in the whole interval; thus we do not need to prove for this case.
Step 2. Assume the function $f(x)$ will have one or more zeros; we can find a zero $f(x_1)=0$ where there will be no other zeros between $x_1$ and $c$.
Step 3. Since $f(x_1)=0, f(c)\ne0$ and the function is continuous, all the function values between $x_1$ and $c$ will have the same sign as $f(c)$ (no more crossing the x-axis in between)
Step 4. Here $x_1$ can be on either side of point $c$; if there are no zeros on the other side, no more proof will be needed.
Step 5. If there is another zero $f(x_2)=0$ on the other side of point $c$, we can use the same arguments above to show that the function values have the same sign as $f(c)$ between $x_2$ and $c$
Step 6. Let $\delta$ be the smaller of $|x_1-c|/2$ and $|x_2-c|/2$; we can conclude that $f(x)$ will have the same sign as $f(c)$ in the interval of $(c-\delta, c+\delta)$
